From 0584ee4aeac04c5f05bcf17652345235b9f927ad Mon Sep 17 00:00:00 2001 From: "W. Trevor King" Date: Mon, 20 May 2013 16:15:57 -0400 Subject: [PATCH] Convert \exp{...} to generate e^{...} Mom wasn't use what exp(...) meant, but she recognized the e^{...} notation. I'm not completely comfortable with how this looks in nested levels during the Gumbel manipulation, but at least it's consistent ;). I may tweak the macro if I change my mind ;). One sneaky bit is the factorials in the \exp{x} nomenclature entry. Searching around, I was bailed out by the interwebs: On 2007-08-21, Nicola Talbot wrote: > ! is a makeindex special character. I haven't used nomencl, but I > expect it requires makeindex. Try "! instead of ! --- src/blurb/gumbel.tex | 18 ++++----- src/cantilever/theory.tex | 2 +- src/local_cmmds.tex | 3 ++ src/sawsim/discussion.tex | 85 ++++++++++++++++++++------------------- src/sawsim/methods.tex | 14 +++++-- 5 files changed, 67 insertions(+), 55 deletions(-) diff --git a/src/blurb/gumbel.tex b/src/blurb/gumbel.tex index 2b9ae38..f868b2a 100644 --- a/src/blurb/gumbel.tex +++ b/src/blurb/gumbel.tex @@ -14,11 +14,11 @@ distribution given by \begin{align} P_\text{GEV}(x|\mu,\sigma,\eta) - &= \frac{1}{\sigma}t(x)\exp\p({-t(x)}) \\ + &= \frac{1}{\sigma}t(x)\exp{-t(x)} \\ t(x) &= \begin{cases} \p({1 + \p({\frac{x-\mu}{\sigma}})\eta})^{-1/\eta} & \text{if } \eta \ne 0 \\ - \exp\p({-\frac{x-\mu}{\sigma}}) & \text{if } \eta = 0 + \exp{-\frac{x-\mu}{\sigma}} & \text{if } \eta = 0 \end{cases} \end{align} where $\mu\in\Reals$ is the location parameter, $\sigma>0$ is the @@ -28,9 +28,9 @@ parameter\citep{wikipedia:GEV}. To recover the Gumbel distribution, set $\eta=0$. \begin{align} P_\text{Gumbel}(x|\mu,\sigma) - &= \frac{1}{\sigma}\exp\p({-\frac{x-\mu}{\sigma}}) - \exp\p({\exp\p({-\frac{x-\mu}{\sigma}})}) \\ - &= \frac{1}{\sigma}\exp\p({z - \exp(-z)}) \;, + &= \frac{1}{\sigma}\exp{-\frac{x-\mu}{\sigma}} + \exp{\exp{-\frac{x-\mu}{\sigma}}} \\ + &= \frac{1}{\sigma}\exp{z - \exp(-z)} \;, \end{align} where $z\equiv (x-\mu)/\sigma$. This form matches \citet{wikipedia:gumbel} and, with the replacements @@ -41,7 +41,7 @@ $\mu\rightarrow\alpha$ and $\sigma\rightarrow\rho$, also matches To recover the Gompertz distribution\citet{wikipedia:gompertz} \begin{align} P_\text{Gompertz}(x|\nu,b) - &= b\nu\exp(bx)\exp(\nu)\exp(-\nu\exp(bx)) + &= b\nu\exp{bx}\exp{\nu}\exp{-\nu\exp{bx}} \end{align} , set $$. @@ -49,8 +49,8 @@ Finally, there are a few other similarly named distributions to watch out for. The Type-1 Gumbel distribution\citet{wikipedia:gumbel-t1} \begin{equation} P_\text{Type-1 Gumbel}(x|a,b) - = ab\exp\p({-(b\exp(-ax)+ax)}) - = -\exp(-b)\cdotP_\text{Gompertz})(x|b,-a) + = ab\exp{-(b\exp{-ax}+ax)}} + = -\exp{-b}\cdotP_\text{Gompertz})(x|b,-a) \end{equation} is similar to the Gompertz distrubution, differing only by a constant scale factor. Since both probability distributions are normalized, @@ -61,7 +61,7 @@ the same as the Gumbel (minimum) distribution. The Type-2 Gumbel distribution\citet{wikipedia:gumbel-t2} \begin{equation} - P_\text{Type-2 Gumbel}(x|a,b) = abx^{-a-1}\exp\p({-bx^{-a}}) + P_\text{Type-2 Gumbel}(x|a,b) = abx^{-a-1}\exp{-bx^{-a}} \end{equation} has $x$ being raised to powers (vs. $e$ being raised to powers in the other distributions), so it is an entirely different beast. diff --git a/src/cantilever/theory.tex b/src/cantilever/theory.tex index a4a9a6c..5bddcb1 100644 --- a/src/cantilever/theory.tex +++ b/src/cantilever/theory.tex @@ -41,7 +41,7 @@ of the applied tension, the energy of the transition state will be where $\kappa$ is the effective linker spring constant for that tension. The Bell-model unfolding rate is thus \begin{align*} - k(f) &= k_0 \exp\p({\frac{f\Delta x - \frac{1}{2}\kappa \Delta x^2}{k_B T}}) \;, + k(f) &= k_0 \exp{\frac{f\Delta x - \frac{1}{2}\kappa \Delta x^2}{k_B T}} \;, \end{align*} and stiffer linkers will increase the mean unfolding force. diff --git a/src/local_cmmds.tex b/src/local_cmmds.tex index ad01d3d..4603e88 100644 --- a/src/local_cmmds.tex +++ b/src/local_cmmds.tex @@ -44,6 +44,9 @@ % Fourier Transform to frequency space \newcommand{\Fourf}[1]{\ensuremath{{\mathcal F}_f\left\{ {#1} \right\}}} +% use e^{...} instead of exp ... +\renewcommand{\exp}[1]{\ensuremath{e^{#1}}} + % Symbol denoting the Langevin function \newcommand{\Langevin}{\ensuremath{\mathcal{L}}} % Symbol denoting big-O order of #1 diff --git a/src/sawsim/discussion.tex b/src/sawsim/discussion.tex index 77e93d0..5a5c7a9 100644 --- a/src/sawsim/discussion.tex +++ b/src/sawsim/discussion.tex @@ -157,8 +157,8 @@ The rate of unfolding events with respect to force is r_{uF} &= -\deriv{F}{N_f} = -\frac{\dd N_f/\dd t}{\dd F/\dd t} = \frac{N_f k_u}{\kappa v} \\ - &= \frac{N_f k_{u0}}{\kappa v}\exp\p({\frac{F\Delta x_u}{k_B T}}) - = \frac{1}{\rho}\exp\p({\frac{F-\alpha}{\rho}}) \;, + &= \frac{N_f k_{u0}}{\kappa v}\exp{\frac{F\Delta x_u}{k_B T}} + = \frac{1}{\rho}\exp{\frac{F-\alpha}{\rho}} \;, \end{align} where $N_f$ is the number of folded domain, $\kappa$ is the spring constant of the cantilever-polymer system, $\kappa v$ is the force @@ -179,9 +179,9 @@ increasing likelihood function follows the Gumbel (minimum) probability density\citep{NIST:gumbel} with $\rho$ and $\alpha$ being the scale and location parameters, respectively\citep{hummer03} \begin{equation} - \mathcal{P}(F) = \frac{1}{\rho} \exp\p[{\frac{F-\alpha}{\rho} - -\exp\p({\frac{F-\alpha}{\rho}}) - }] \;. \label{eq:sawsim:gumbel} + \mathcal{P}(F) = \frac{1}{\rho} \exp{\frac{F-\alpha}{\rho} + -\exp{\frac{F-\alpha}{\rho}} + } \;. \label{eq:sawsim:gumbel} \end{equation} The distribution has a mean $\avg{F}=\alpha-\gamma_e\rho$ and a variance $\sigma^2 = \pi^2\rho^2/6$, where $\gamma_e=0.577\ldots$ is @@ -691,7 +691,7 @@ $N_f$ in terms of $k_u$ as follows: k_u &\equiv -\frac{1}{N_f} \deriv{t}{N_f} \\ -k_u \dd t \cdot \deriv{t}{F} &= \frac{\dd N_f}{N_f} \\ \frac{-1}{\kappa v} \int k_0 \dd F &= \ln(N_f) + c \\ - N_f &= C\exp{\p({\frac{-1}{\kappa v}\integral{}{}{F}{k_u}})} \;, + N_f &= C\exp{\frac{-1}{\kappa v}\integral{}{}{F}{k_u}} \;, \label{eq:N_f} \end{align} where $c \equiv \ln(C)$ is a constant of integration scaling $N_f$. @@ -700,12 +700,12 @@ where $c \equiv \ln(C)$ is a constant of integration scaling $N_f$. 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 \\ + P &= C\exp{\frac{-1}{kv}\integral{}{}{F}{\kappa}} + = C\exp{\frac{-1}{kv}\kappa F} + = C\exp{\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}})} + = \frac{W\kappa P_0}{vk} \exp{\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. @@ -715,25 +715,28 @@ Not the most earth shattering result, but it's a comforting first step, and it d 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) \;, + \kappa = \kappa_0 \cdot \exp{\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)]}\} \\ + P &= C\exp{\frac{-1}{kv}\integral{}{}{F}{\kappa}} + = C\exp{\frac{-1}{kv} \frac{\kappa_0}{a} \exp{a F}} + = C\exp{\frac{-\kappa_0}{akv}\exp{a F}} \\ + P(0) &\equiv P_0 = C\exp{\frac{-\kappa_0}{akv}} \\ + C &= P_0 \exp{\frac{\kappa_0}{akv}} \\ + P &= P_0 \exp{\frac{\kappa_0}{akv}\p({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}\;. + = \frac{W}{vk} P_0 + \exp{\frac{\kappa_0}{akv}\p({1-\exp{a F}})} \kappa_0 \exp{a F} + = \frac{W\kappa_0 P_0}{vk} + \exp{a F + \frac{\kappa_0}{akv}\p({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)}] \;, + h(F) \propto \exp{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. @@ -741,12 +744,12 @@ 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}}) \;, + p(x) &\propto z\exp{-z} \\ + z &\equiv \exp{-\frac{x-\mu}{\beta}} \;, \end{align} but we have \begin{equation} - p(x) \propto z\exp(-bz) \;. + 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 @@ -757,15 +760,15 @@ 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') \;, + 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}}) - }] \label{eq:sawsim:gumbel-x} + P(x) = \frac{1}{\beta} \exp{\frac{x-\alpha}{\beta} + -\exp{\frac{x-\alpha}{\beta}}} + \label{eq:sawsim:gumbel-x} \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 @@ -775,18 +778,16 @@ $\alpha=-\beta\ln(\kappa\beta/kv)$, and $F=x$ we have distribution, \cref{eq:sawsim:gumbel-x}).} \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)] + &= \frac{1}{\beta} \exp{\frac{F+\beta\ln(\kappa\beta/kv)}{\beta} + -\exp{\frac{F+\beta\ln(\kappa\beta/kv)}{\beta}}} \\ + &= \frac{1}{\beta} \exp{F/\beta}\exp{\ln(\kappa\beta/kv)} + \exp{-\exp{F/\beta}\exp{\ln(\kappa\beta/kv)}} \\ + &= \frac{1}{\beta} \frac{\kappa\beta}{kv} \exp{F/\beta} + \exp{-\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 @@ -799,7 +800,7 @@ constant loading does indeed follow the Gumbel distribution. 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} - k_u = \frac{D}{l_b l_{ts}} \cdot \exp\p({\frac{-U_b(F)}{k_B T}}) \;, + k_u = \frac{D}{l_b l_{ts}} \cdot \exp{\frac{-U_b(F)}{k_B T}} \;, \label{eq:kramers-saddle} \end{equation} where $U_b(F)$ is the barrier height under an external force $F$, diff --git a/src/sawsim/methods.tex b/src/sawsim/methods.tex index ab3a05e..2ec8891 100644 --- a/src/sawsim/methods.tex +++ b/src/sawsim/methods.tex @@ -405,11 +405,19 @@ According to the theory developed by \citet{bell78} and extended by unfolding rate constant of a protein molecule \index{Bell model} \begin{equation} - k_u = k_{u0} \exp\p({\frac{F\Delta x_u}{k_B T}}) \;, \label{eq:sawsim:bell} + k_u = k_{u0} \exp{\frac{F\Delta x_u}{k_B T}} \;, \label{eq:sawsim:bell} \end{equation} where $k_{u0}$ is the unfolding rate in the absence of an external 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, + \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, $e=2.718\ldots$.} \begin{figure} \asyinclude{figures/schematic/landscape-bell} @@ -494,9 +502,9 @@ proteins with broad free energy barriers. \frac{1}{k_u} = \frac{1}{D} \integral{-\infty}{\infty}{x}{% - e^{\frac{U_F(x)}{k_B T}} + \exp{\frac{U_F(x)}{k_B T}} \integral{-\infty}{x}{x'}{% - e^{\frac{-U_F(x')}{k_B T}}}} \;, + \exp{\frac{-U_F(x')}{k_B T}}}} \;, \label{eq:kramers} \end{equation} where $D$ is the diffusion coefficient and $U_F(x)$ is the free energy -- 2.26.2