-SUBDIRS = packages sawsim cantilever_calib
+SUBDIRS = packages sawsim unfolding_distributions cantilever_calib
all :
@for i in $(SUBDIRS); do \
\end{acknowledgments}
\tableofcontents
-% Include these following commands only if you have tables or figures.
-% Tables should come before figures!
\listoftables
\listoffigures
\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}
\linenumbers
-
\chapter{Monte Carlo mechanical unfolding simulation}
\section{Introduction}
--- /dev/null
+\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}
+
--- /dev/null
+\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.
--- /dev/null
+\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}
--- /dev/null
+\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}).
--- /dev/null
+\linenumbers
+\chapter{Theoretical unfolding force distributions}
+\input{unfolding_distributions/overview}
+\input{unfolding_distributions/review}
+\input{unfolding_distributions/singledomain_constantloading}
+\input{unfolding_distributions/kramers}
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",
}
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.",
}
projects / thesis.git / commitdiff