From ba274effaa00df071b23ac2f153777562379b6e1 Mon Sep 17 00:00:00 2001 From: "W. Trevor King" Date: Thu, 14 Jan 2010 12:19:01 -0500 Subject: [PATCH] Brought in unfolding_distributions --- tex/src/Makefile | 2 +- tex/src/root.tex | 5 +- tex/src/sawsim/sawsim.tex | 1 - tex/src/unfolding_distributions/kramers.tex | 20 +++ tex/src/unfolding_distributions/overview.tex | 6 + tex/src/unfolding_distributions/review.tex | 85 +++++++++ .../singledomain_constantloading.tex | 167 ++++++++++++++++++ .../unfolding_distributions.tex | 6 + tex/src/wtk.bib | 4 +- 9 files changed, 290 insertions(+), 6 deletions(-) create mode 100644 tex/src/unfolding_distributions/kramers.tex create mode 100644 tex/src/unfolding_distributions/overview.tex create mode 100644 tex/src/unfolding_distributions/review.tex create mode 100644 tex/src/unfolding_distributions/singledomain_constantloading.tex create mode 100644 tex/src/unfolding_distributions/unfolding_distributions.tex diff --git a/tex/src/Makefile b/tex/src/Makefile index b40dbde..3fa6f12 100644 --- a/tex/src/Makefile +++ b/tex/src/Makefile @@ -1,4 +1,4 @@ -SUBDIRS = packages sawsim cantilever_calib +SUBDIRS = packages sawsim unfolding_distributions cantilever_calib all : @for i in $(SUBDIRS); do \ diff --git a/tex/src/root.tex b/tex/src/root.tex index b932459..b9f76ce 100644 --- a/tex/src/root.tex +++ b/tex/src/root.tex @@ -30,8 +30,6 @@ R01-GM071793. \end{acknowledgments} \tableofcontents -% Include these following commands only if you have tables or figures. -% Tables should come before figures! \listoftables \listoffigures @@ -41,10 +39,13 @@ R01-GM071793. \end{preamble} \begin{thesis} +\pdfbookmark[-1]{Mainmatter}{Mainmatter} +\include{unfolding_distributions/unfolding_distributions} \include{sawsim/sawsim} \end{thesis} \bibliography{% +% unfolding_distributions/unfolding_distributions, % sawsim/sawsim,% currently empty cantilever_calib/cantilever_calib,% wtk} diff --git a/tex/src/sawsim/sawsim.tex b/tex/src/sawsim/sawsim.tex index f1bd4b7..34ac919 100644 --- a/tex/src/sawsim/sawsim.tex +++ b/tex/src/sawsim/sawsim.tex @@ -1,5 +1,4 @@ \linenumbers - \chapter{Monte Carlo mechanical unfolding simulation} \section{Introduction} diff --git a/tex/src/unfolding_distributions/kramers.tex b/tex/src/unfolding_distributions/kramers.tex new file mode 100644 index 0000000..47ad47d --- /dev/null +++ b/tex/src/unfolding_distributions/kramers.tex @@ -0,0 +1,20 @@ +\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, 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 floating around in a finished 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/tex/src/unfolding_distributions/overview.tex b/tex/src/unfolding_distributions/overview.tex new file mode 100644 index 0000000..4c10a27 --- /dev/null +++ b/tex/src/unfolding_distributions/overview.tex @@ -0,0 +1,6 @@ +\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/tex/src/unfolding_distributions/review.tex b/tex/src/unfolding_distributions/review.tex new file mode 100644 index 0000000..aa8b601 --- /dev/null +++ b/tex/src/unfolding_distributions/review.tex @@ -0,0 +1,85 @@ +\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-vazquez99b,schlierf06} due to it's simplicity and ease of use\citep{hummer03}. +Kramers introduced his theory in the context of thermally activated barrier crossing, 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). +Early work on mechanical unfolding focused on \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 has 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} introduces 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/tex/src/unfolding_distributions/singledomain_constantloading.tex b/tex/src/unfolding_distributions/singledomain_constantloading.tex new file mode 100644 index 0000000..20e4aae --- /dev/null +++ b/tex/src/unfolding_distributions/singledomain_constantloading.tex @@ -0,0 +1,167 @@ +\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} +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} +Suprise! 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 an awful lot like the 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). + +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 gets funky with diffusion constants that depend on the +protein's end to end distance, and I haven't worked out the math +yet\ldots + + +\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}). diff --git a/tex/src/unfolding_distributions/unfolding_distributions.tex b/tex/src/unfolding_distributions/unfolding_distributions.tex new file mode 100644 index 0000000..e34172b --- /dev/null +++ b/tex/src/unfolding_distributions/unfolding_distributions.tex @@ -0,0 +1,6 @@ +\linenumbers +\chapter{Theoretical unfolding force distributions} +\input{unfolding_distributions/overview} +\input{unfolding_distributions/review} +\input{unfolding_distributions/singledomain_constantloading} +\input{unfolding_distributions/kramers} diff --git a/tex/src/wtk.bib b/tex/src/wtk.bib index bc15c40..66f3283 100644 --- a/tex/src/wtk.bib +++ b/tex/src/wtk.bib @@ -311,7 +311,7 @@ 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 Hanggi Eq. 4.56c (page 275)\cite{hanggi90}.", + note = "Derives the major theory behind my thesis. The Kramers rate equation is H{\"a}nggi Eq. 4.56c (page 275)\cite{hanggi90}.", project = "Energy Landscape Roughness", } @@ -2247,7 +2247,7 @@ 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). $\nu \equiv k$ is the force/time-dependent off rate... (TODO) - The Kramers' rate equation (second equation in the paper) is Hanggi Eq.~4.56b (page 275)\cite{hanggi90}. + The Kramers' rate equation (second equation in the paper) is H{\"a}nggi Eq.~4.56b (page 275)\cite{hanggi90}. It is important to extract $k_0$ and $\Delta x$ using every available method.", } -- 2.26.2