(AthenaES) in PBS on gold-coated coverslips. We used both cantilevers
on Olympus's OMCL-TR400-PSA-1, which are nominally $80$ and
$20\U{pN/nm}$. Promising sawtooth curves were selected by eye and fit
-to WLCs to identify I27 unfolding events. The results were sorted
-into two bins according to cantilever stiffness, and then averaged
-across each cantilever-stiffness/pulling-speed group to produce the
-following graph.
+to WLCs\index{WLC} to identify I27 unfolding events. The results were
+sorted into two bins according to cantilever stiffness, and then
+averaged across each cantilever-stiffness/pulling-speed group to
+produce the following graph.
\begin{figure}
\includegraphics[width=4in]{figures/cantilever-data/plot_splits}
mean unfolding force is not entirely due to the increased loading rate
of the stiffer cantilever, because the difference is still present in
the loading rate dependence. The loading rates were extracted from
-the data by taking the slope of the fit WLC at unfolding.
+the data by taking the slope of the fit WLC\index{WLC} at unfolding.
\begin{figure}
\asyfig{figures/cantilever-data/loading-rate}
We can model the I27 multimers as an array of Bell-model unfolders in
series with a cantilever. Any unfolded domains also contribute to the
-tension according to the WLC tension formula. Completing 1000
-simulated pulls for each cantilever/pulling-speed/multimer-number
-combination with our \sawsim Monte Carlo simulator yielded the
+tension according to the WLC\index{WLC} tension formula. Completing
+1000 simulated pulls for each cantilever/pulling-speed/multimer-number
+combination with our \sawsim\ Monte Carlo simulator yielded the
following results
(\cref{fig:cant:sim:v-dep,fig:cant:sim:load-dep,fig:cant:sim:i-dep}).
\end{align*}
and stiffer linkers will increase the mean unfolding force.
-Unfolded I27 domains can be well-modeled as wormlike-chains
-(WLCs)\cite{carrion-vazquez99b}, producing a tension under extension
-approximated by Bustamante's interpolation
-formula\cite{bustamante94}.%,marko95}.
-\begin{align*}
- F_\text{WLC} &= \frac{k_B T}{p}\p[{\frac{1}{4}\p({\frac{1}{(1-x/L)^2}-1})+\frac{x}{L}}] \;,
-\end{align*}
-where $p \approx 4\U{\AA}$ is the persistence length, and $L \approx
-28\U{nm}$ is the contour length of the unfolded domain. Obviously
-effective stiffness of an unfolded I27 domain is highly dependent on
-the unfolding force, and for tensions $\sim 280\U{pN}$ is $\sim
-190\U{pN/nm}$. This is within a factor of four of common cantilever
-spring constants, so cantilever stiffness drives the effective spring
-constant for the first four domains, after which point I27 stiffness
-takes the lead.
+Unfolded I27 domains can be well-modeled as wormlike chains (WLCs,
+\cref{sec:tension:wlc})\cite{carrion-vazquez99b}, where $p \approx
+4\U{\AA}$ is the persistence length, and $L \approx 28\U{nm}$ is the
+contour length of the unfolded domain. Obviously effective stiffness
+of an unfolded I27 domain is highly dependent on the unfolding force,
+and for tensions $\sim 280\U{pN}$ is $\sim 190\U{pN/nm}$. This is
+within a factor of four of common cantilever spring constants, so
+cantilever stiffness drives the effective spring constant for the
+first four domains, after which point I27 stiffness takes the lead.
\begin{figure}
\begin{tikzpicture}
% Inspired by Florian Hollandt's RNA codons
deflection, $x_u$ is the extension of the unfolded polymer, and
$x_f=x_{f1}+x_{f2}$ is the extension of the folded polymer. (b)
An experimental force curve from stretching a ubiquitin polymer
- with the rising parts of the peaks fitted to the WLC
- model\citep{chyan04}. The pulling speed used was $1\U{$\mu$m/s}$.
- The irregular features at the beginning of the curve are due to
- nonspecific interactions between the tip and the substrate
- surface, and the last high force peak is caused by the detachment
- of the polymer from the tip or the substrate surface. Note that
- the abscissa is the extension of the protein chain $x_t-x_c$.}
+ with the rising parts of the peaks fitted to the WLC\index{WLC}
+ model (\cref{sec:tension:wlc})\citep{chyan04}. The pulling speed
+ used was $1\U{$\mu$m/s}$. The irregular features at the beginning
+ of the curve are due to nonspecific interactions between the tip
+ and the substrate surface, and the last high force peak is caused
+ by the detachment of the polymer from the tip or the substrate
+ surface. Note that the abscissa is the extension of the protein
+ chain $x_t-x_c$.}
\end{center}
\end{figure}
\end{equation}
where $N_f$ and $\kappa$ depend on the domain index $i=N_u$. Curves based
on this formula fit the simulated data remarkably well considering the
-effective WLC stiffness $\kappa_\text{WLC}$ is the only fitted
+effective WLC\index{WLC} stiffness $\kappa_\text{WLC}$ is the only fitted
parameter, and that the actual WLC stiffness is not constant, as we
have assumed here, but a non-linear function of $F$.
\end{center}
\end{figure}
-\subsection{The effect of polymer inhomogeneity}
-\label{sec:sawsim:results-folded-tension}
-
-The unfolded polypeptide chain has been shown to follow the WLC model
-quite well, though other polymer models, such as the Freely-Jointed
-Chain (FJC)\citep{verdier70}, can be used to fit the force-extension
-relationship\citep{janshoff00}. A short chain of folded proteins,
-however, cannot be described well by polymer models. Several studies
-have used WLC and FJC models to fit the elastic properties of the
-modular protein titin\citep{granzier97,linke98a},
-% TODO: check it really is folded domains \& bulk titin
-but native titin contains hundreds of folded and unfolded domains
-domains. For the short protein polymers common in mechanical
-unfolding experiments, the cantilever dominates the elasticity of the
-polymer-cantilever system before any protein molecules unfold. After
-the first unfolding event occurs, the unfolded portion of the chain is
-already longer and softer than the sum of all the remaining folded
-domains, and dominates the elasticity of the whole chain. Therefore,
-the details of the tension model chosen for the folded domains has
-negligible effect on the unfolding forces, which was also suggested by
-\citet{staple08}. Force curves simulated using different models to
-describe the folded domains yielded almost identical unfolding force
-distributions (data not shown).
\subsection{The effect of cantilever force constant}
\label{sec:sawsim:cantilever}
each unfolding event is not equal to the distance between adjacent
peaks in the force curve because the chain is never fully stretched.
This contour length increase can only be obtained by fitting the curve
-to WLC or other polymer models (\cref{fig:expt-sawtooth}).
+to WLC\index{WLC} or other polymer models (\cref{fig:expt-sawtooth}).
\begin{figure}
\begin{center}
F = \kappa_c x_c \;, \label{eq:sawsim:hooke}
\end{equation}
where $\kappa_c$ is the bending spring constant and $x_c$ is the
-deflection of the cantilever (\cref{fig:unfolding-schematic}). Unfolded domains
-are modeled as a Worm-Like Chain (WLC)\citep{marko95,bustamante94}, in
-which the tension $F$ is related to its extension (end-to-end
-distance) $x_u$ by
-\begin{equation}
- F = \frac{k_B T}{p_u}
- \p[{ \frac{1}{4}\p({ \frac{1}{(1-x_u/L_u)^2} - 1 })
- + \frac{x_u}{L_u} }] \;,
- \label{eq:sawsim:wlc}
-\end{equation}
-where $p_u$ is the persistence length and $L_u=N_uL_{u1}$ is the total
-contour length of the unfolded chain. The chain of $N_f$ folded
-domains is modeled as a string free to assume any extension up to some
-fixed contour length $L_f=N_fL_{f1}$
+deflection of the cantilever (\cref{fig:unfolding-schematic}).
+Unfolded domains are modeled as WLCs (\cref{sec:tension:wlc}).
+
+The chain of $N_f$ folded domains is modeled as a string free to
+assume any extension up to some fixed contour length $L_f=N_fL_{f1}$
\begin{equation}
F = \begin{cases}
0 & \text{if $x_f<L_f$} \;, \\
The tension calculation assumes an equilibrated chain, so
consideration must be given to the chain's relaxation time, which
should be short compared to the loading timescale. The relaxation
-time for a WLC is given by
+time for a WLC\index{WLC!relaxation time} is given by
\begin{equation}
\tau \approx \eta \frac{k_BT p}{F^2}
% < \eta \frac{k_BT p}{(k_BTx/pL)^2} =
--- /dev/null
+\section{Folded domain tension}
+\label{sec:tension:folded}
+
+The unfolded polypeptide chain has been shown to follow the
+WLC\index{WLC} model quite well (\cref{sec:tension:wlc}), though other
+polymer models, such as the Freely-Jointed Chain
+(FJC)\citep{verdier70}\index{FJC}\nomenclature{FJC}{Freely-Jointed Chain}
+(\cref{sec:tension:fjc}), can be used to fit the force-extension
+relationship\citep{janshoff00}. A short chain of folded proteins,
+however, cannot be described well by polymer models. Several studies
+have used WLC and FJC models to fit the elastic properties of the
+modular protein titin\citep{granzier97,linke98a},
+% TODO: check it really is folded domains \& bulk titin
+but native titin contains hundreds of folded and unfolded domains
+domains. For the short protein polymers common in mechanical
+unfolding experiments, the cantilever dominates the elasticity of the
+polymer-cantilever system before any protein molecules unfold. After
+the first unfolding event occurs, the unfolded portion of the chain is
+already longer and softer than the sum of all the remaining folded
+domains, and dominates the elasticity of the whole chain. Therefore,
+the details of the tension model chosen for the folded domains has
+negligible effect on the unfolding forces, which was also suggested by
+\citet{staple08}. Force curves simulated using different models to
+describe the folded domains yielded almost identical unfolding force
+distributions (data not shown).
\chapter{Chain Tension}
\label{sec:tension}
+
+\input{tension/polymer}
+\input{tension/folded}
--- /dev/null
+\section{Polymer Models}
+
+
+\subsection{Worm-like chains}
+\label{sec:tension:wlc}
+
+The unfolded forms of many domains can be modeled as Worm-Like Chains
+(WLCs)\citep{marko95,bustamante94}
+\index{WLC|textbf}\nomenclature{WLC}{Wormlike Chain}, which treats the
+unfolded polymer as an elastic rod of persistence length $p$ and
+contour length $L$. The relationship between tension $F$ and
+extension (end-to-end distance) $x$ is given to within XX\% by
+Bustamante's interpolation formula\citep{marko95,bustamante94}.
+\begin{equation}
+ F_\text{WLC}(x,p,L) = \frac{k_B T}{p_u}
+ \p[{ \frac{1}{4}\p({ \frac{1}{(1-x/L)^2} - 1 })
+ + \frac{x}{L} }] \;,
+ \label{eq:sawsim:wlc}
+\end{equation}
+where $p$ is the persistence length.
+
+For chain with $N_u$ unfolded domains sharing a persistence length
+$p_u$ and per-domain contour lengths $L_{u1}$, the tension of the WLC
+is determine by summing the contour lengths
+\begin{equation}
+ F(x, p_u, L_u, N_u) = F_\text{WLC}(x, p_u, N_uL_{u1})
+\end{equation}
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.
+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 floating around in a finished form. However, we do have
\section{Overview}
-For testing the |sawsim| program, we need a few analytic solutions to unfolding distributions.
+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.
So our unfolding force histogram for a single Bell domain under
constant loading does indeed follow the Gumbel distribution.
-\subsection{Saddle-point Kramers model}
+\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).
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