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-\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}
+++ /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}
-\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}).
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