From: W. Trevor King Date: Sat, 14 May 2011 19:02:07 +0000 (-0400) Subject: Remove src/unfolding-distributions/* which were moved to src/unfolding/distributions-*. X-Git-Tag: v1.0~353 X-Git-Url: http://git.tremily.us/?a=commitdiff_plain;h=2f291bd8594645727f0b79e3290ce5d1fbec4bf2;p=thesis.git Remove src/unfolding-distributions/* which were moved to src/unfolding/distributions-*. --- diff --git a/src/unfolding-distributions/kramers.tex b/src/unfolding-distributions/kramers.tex deleted file mode 100644 index 55ae7bf..0000000 --- a/src/unfolding-distributions/kramers.tex +++ /dev/null @@ -1,19 +0,0 @@ -\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} - diff --git a/src/unfolding-distributions/main.tex b/src/unfolding-distributions/main.tex deleted file mode 100644 index 22481c6..0000000 --- a/src/unfolding-distributions/main.tex +++ /dev/null @@ -1,7 +0,0 @@ -\chapter{Theoretical unfolding force distributions} -\label{sec:unfolding-distributions} - -\input{unfolding-distributions/overview} -\input{unfolding-distributions/review} -\input{unfolding-distributions/singledomain_constantloading} -\input{unfolding-distributions/kramers} diff --git a/src/unfolding-distributions/overview.tex b/src/unfolding-distributions/overview.tex deleted file mode 100644 index abddc1f..0000000 --- a/src/unfolding-distributions/overview.tex +++ /dev/null @@ -1,6 +0,0 @@ -\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. diff --git a/src/unfolding-distributions/review.tex b/src/unfolding-distributions/review.tex deleted file mode 100644 index 42b2540..0000000 --- a/src/unfolding-distributions/review.tex +++ /dev/null @@ -1,85 +0,0 @@ -\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{rief97b,carrion-vazquez99a,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{rief97b}.%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{rief97b}: -\end{itemize} - -\subsection{History of experimental laser tweezer unfolding experiments} - -\begin{itemize} - \item \citet{izrailev97}: -\end{itemize} diff --git a/src/unfolding-distributions/singledomain_constantloading.tex b/src/unfolding-distributions/singledomain_constantloading.tex deleted file mode 100644 index 902896e..0000000 --- a/src/unfolding-distributions/singledomain_constantloading.tex +++ /dev/null @@ -1,169 +0,0 @@ -\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}).