&= \frac{G_0 \pi}{2m \beta k}
\end{align}
The integration is detailed in \cref{sec:integrals}.
-By our corollary to Parseval's theorem (\cref{eq:parseval-var}), we have
+By the corollary to Parseval's theorem (\cref{eq:parseval-var}), we have
\begin{equation}
\avg{x(t)^2} = \frac{G_0 \pi}{2m^2\beta\omega_0^2} \label{eq:DHO-var}
\end{equation}
The experiments were carried out on octomers of I27 (\cref{fig:I27}).
I27 is a model protein that has been used in mechanical unfolding
experiments since the first use of synthetic
-chains\citep{carrion-vazquez99b,TODO}. We use it here because it is
+chains\citep{carrion-vazquez99b,TODO}. It was used here because it is
both well characterized and readily available (%
\href{http://www.athenaes.com/}{AthenaES}, Baltimore, MD,
\href{http://www.athenaes.com/I27OAFMReferenceProtein.php}{0304}).
\label{fig:I27}}
\end{figure}
-The I27 octamers were stored in a ... buffer solution.
+The I27 octamers were stored in a TODO buffer solution.
Mechanical unfolding experiments were carried out on I27 octomers
(AthenaES) in PBS on gold-coated coverslips. We used both cantilevers
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.
+produce \cref{fig:plot-splits}.
\begin{figure}
\includegraphics[width=4in]{figures/cantilever-data/plot_splits}
- \caption{plot splits.}
+ \caption{plot splits.\label{fig:plot-splits}}
\end{figure}
Unfortunately, the data are not of high enough quality to extract the
\caption{Energy landscape schematic.\label{fig:landscape}}
\end{figure}
-The presence of attached linkers and cantilever alters the free energy
-landscape. Tension in the linkers favors domain unfolding, but that
-tension is not necessarily independent of the unfolding reaction
+The presence of attached linkers and cantilevers alters the free
+energy landscape. Tension in the linkers favors domain unfolding, but
+that tension is not necessarily independent of the unfolding reaction
coordinate. For sufficiently stiff cantilevers and linkers, even the
-small extension of the domain as it shifts from it's bound to
+small extension of the domain as it shifts from its bound to
transition state noticeably reduces the effective tension. Assuming
the bound and transition state extensions are relatively independent
of the applied tension, the energy of the transition state will be
\begin{figure}
\begin{center}
\includegraphics[width=2in]{figures/biotin-streptavidin/1SWE.png}%
- \caption{Complex of biotin (red) and a streptavidin tetramer (green)
+ \caption{Complex of biotin\index{biotin} (red) and a
+ streptavidin\index{streptavidin} tetramer (green)
(\href{http://dx.doi.org/10.2210/pdb1swe/pdb}{PDB ID: 1SWE})%
\citep{freitag97}. The correct streptavidin conformation creates
- the biotin-specific binding pockets. Biotin-streptavidin is a model
- ligand-receptor pair isolated from the bacterium
- \species{Streptomyces avidinii}. Streptavidin binds to cell
- surfaces, and bound biotin increases streptavidin's binding
- affinity\citep{alon90}.
- Figure generated with \citetalias{pymol}.
+ the biotin-specific binding pockets. Biotin-streptavidin is a
+ model ligand-receptor pair isolated from the bacterium
+ \species{Streptomyces avidinii}%
+ \index{Streptomyces@\species{Streptomyces avidnii}}. Streptavidin
+ binds to cell surfaces, and bound biotin increases streptavidin's
+ cell-binding affinity\citep{alon90}. Figure generated with
+ \citetalias{pymol}.
\label{fig:ligand-receptor}}
\end{center}
\end{figure}
\subfloat[][]{\includegraphics[width=2in]{figures/schematic/dill97-fig4}%
\label{fig:folding:landscape}}
\caption{(a) A ``double T'' example of the pathway model of protein
- folding, in which the protein proceeds through a series of
- metastable transition states $I_1$ and $I_2$ with two ``dead end''
- states $I_1^X$ and $I_2^X$. Adapted from \citet{bedard08}. (b)
- The landscape model of protein folding, in which the protein
- diffuses through a multi-dimensional free energy landscape.
- Separate folding attempts may take many distinct routes through
- this landscape on the way to the folded state. Reproduced from
- \citet{dill97}.
+ folding, in which the protein proceeds from the native state $N$
+ to the unfolded state $U$ via a series of metastable transition
+ states $I_1$ and $I_2$ with two ``dead end'' states $I_1^X$ and
+ $I_2^X$. Adapted from \citet{bedard08}. (b) The landscape model
+ of protein folding, in which the protein diffuses through a
+ multi-dimensional free energy landscape. Separate folding
+ attempts may take many distinct routes through this landscape on
+ the way to the folded state. Reproduced from \citet{dill97}.
\label{fig:folding}}
\end{center}
\end{figure}
\section{Mechanical unfolding experiments}
-% AFM unfolding procedure
-In a mechanical unfolding experiment, a protein polymer is tethered
-between two surfaces: a flat substrate and an AFM tip. The polymer is
-stretched by increasing the separation between the two surfaces
-(\cref{fig:unfolding-schematic}). The most common mode is the
-constant speed experiment in which the substrate surface is moved away
-from the tip at a uniform rate. The tethering surfaces, \ie, the AFM
-tip and the substrate, have much larger radii of curvature than the
-dimensions of single domain globular proteins that are normally used
-for folding studies. This causes difficulties in manipulating
-individual protein molecules because nonspecific interactions between
-the AFM tip and the substrate may be stronger than the forces required
-to unfold the protein when the surfaces are a few nanometers apart.
-To circumvent these difficulties, globular protein molecules are
-linked into polymers, which are then used in the AFM
-studies\citep{carrion-vazquez99b,chyan04,carrion-vazquez03}. When
+% AFM unfolding procedure In a mechanical unfolding experiment, a
+protein polymer is tethered between two surfaces: a flat substrate and
+an AFM tip. The polymer is stretched by increasing the separation
+between the two surfaces (\cref{fig:unfolding-schematic}). The most
+common mode is the constant speed experiment in which the substrate
+surface is moved away from the tip at a uniform rate. The tethering
+surfaces, \ie, the AFM tip and the substrate, have much larger radii
+of curvature than the dimensions of single domain globular proteins
+that are normally used for folding studies. This causes difficulties
+in manipulating individual protein molecules because nonspecific
+interactions between the AFM tip and the substrate may be stronger
+than the forces required to unfold the protein when the surfaces are a
+few nanometers apart. To circumvent these difficulties, globular
+protein molecules are linked into polymers, which are then used in the
+AFM studies\citep{carrion-vazquez99b,chyan04,carrion-vazquez03}. When
such a polymer is pulled from its ends, each protein molecule feels
the externally applied force, which increases the probability of
unfolding by reducing the free energy barrier between the native and
occurring immediately. The force versus extension relationship, or
\emph{force curve}, shows a typical sawtooth pattern
(\cref{fig:expt-sawtooth}), where each peak corresponds to the
-unfolding of a single protein in the polymer. Therefore, the
+unfolding of a single protein domain in the polymer. Therefore, the
individual unfolding events are separated from each other in space and
-time, facilitating single molecule studies.
+time, allowing single molecule resolution despite the use of
+multi-domain test proteins.
\begin{figure}
\begin{center}
mechanical unfolding experiments. This theory makes straightforward
analysis of unfolding results difficult, so \cref{sec:sawsim} presents
a Monte Carlo simulation approach to fitting unfolding parameters, and
-\cref{sec:contour-space} presents the contour-length space approach to
-fingerprinting unfolding pathways. \Cref{sec:temperature-theory}
-wraps up the theory section by extending the analysis in
-\cref{sec:unfolding,sec:unfolding-distributions} to multiple
-temperatures.
+\cref{sec:contour-space} presents the contour-length space analysis
+for converting force curves to unfolding pathway fingerprints.
+\Cref{sec:temperature-theory} wraps up the theory section by extending
+the analysis in \cref{sec:unfolding,sec:unfolding-distributions} to
+multiple temperatures.
\Cref{sec:apparatus} describes our experimental apparatus and methods,
as well as calibration procedures. With both the theory and procedure
+% Remove the \leftmark (chapter title), since some chapter/section
+% titles would overlap.
+\fancyfoot[RE,LO]{}
+
\usepackage[super,sort&compress]{natbib} % fancy citation extensions
% super selects citations in superscript mode
% sort&compress automatically sorts and compresses compound citations (\cite{a,b,...})
doi = "10.1038/sj.embor.7400403",
URL = "http://www.nature.com/embor/journal/v6/n5/abs/7400403.html",
eprint = "http://www.nature.com/embor/journal/v6/n5/pdf/7400403.pdf",
- note = "Applies H&T\cite{hyeon03} to ligand-receptor
+ note = "Applies H\&T\cite{hyeon03} to ligand-receptor
binding.",
project = "Energy Landscape Roughness",
}
doi = "10.1073/pnas.120048697",
URL = "http://www.pnas.org/cgi/content/abstract/97/12/6527",
eprint = "http://www.pnas.org/cgi/reprint/97/12/6527.pdf",
- note = "Unfolding order not from protein-surface interactions. XOXOXO... experiment.",
+ note = "Unfolding order not from protein-surface interactions. Mechanical unfolding of a chain of interleaved domains $ABABAB\ldots$ yielded a run of $A$ unfoldings followed by a run of $B$ unfoldings.",
}
@Article{nome07,
}
@Article{hanggi90,
- title = {Reaction-rate theory: fifty years after Kramers},
+ title = {Reaction-rate theory: fifty years after {K}ramers},
author = {H\"anggi, Peter and Talkner, Peter and Borkovec, Michal },
journal = {Rev. Mod. Phys.},
volume = {62},
@Article{olshansky97,
author = "S. J. Olshansky and B. A. Carnes",
- title = "Ever since Gompertz",
+ title = "Ever since {G}ompertz",
journal = "Demography",
year = "1997",
month = feb,
@Article{juckett93,
author = "D. A. Juckett and B. Rosenberg",
- title = "Comparison of the Gompertz and Weibull functions as
+ title = "Comparison of the {G}ompertz and {W}eibull functions as
descriptors for human mortality distributions and their
intersections",
journal = "Mech Ageing Dev",
- year = "1993",
+ year = 1993,
month = jun,
- volume = "69",
- number = "1-2",
+ volume = 69,
+ number = "1--2",
pages = "1--31",
keywords = "Adolescent",
keywords = "Adult",
@Article{walton08,
author = "Emily B. Walton and Sunyoung Lee and Krystyn J. {Van
Vliet}",
- title = "Extending Bell's model: how force transducer stiffness
+ title = "Extending {B}ell's model: how force transducer stiffness
alters measured unbinding forces and kinetics of
molecular complexes",
journal = BPJ,
url = "http://stacks.iop.org/0953-8984/8/7561",
eprint = "http://www.iop.org/EJ/article/0953-8984/8/41/006/c64103.pdf",
project = "Cantilever Calibration",
- note = "Actually write down Lagrangian formula and give a decent
- derivation of PSD. Don't show or work out the integrals
- though...",
+ note = "They actually write down Lagrangian formula and give a decent
+ derivation of PSD, but don't show or work out the integrals.",
}
@Article{staple08,
spring constant $\kappa_c=50\U{pN/nm}$, temperature $T=300\U{K}$,
persistence length of unfolded proteins $p_u=0.40\U{nm}$, $\Delta
x_u=0.225\U{nm}$, and $k_{u0}=5\E{-5}\U{s$^{-1}$}$. The contour
- length between the two linking point on a protein molecule is
+ length between the two linking points on a protein molecule is
$L_{f1}=3.7\U{nm}$ in the folded form and $L_{u1}=28.1\U{nm}$ in the
unfolded form. These parameters are those of ubiquitin molecules
connected through the N-C termini\citep{chyan04,carrion-vazquez03}.
surface often cause irregular features in the beginning of the force
curve (\cref{fig:expt-sawtooth}), making the identification of the
first peak uncertain. Furthermore, it is often difficult to acquire a
-large amount of data in single molecules experiments. These
+large amount of data in single molecule experiments. These
difficulties make the aforementioned data analysis approach unfeasible
for many mechanical unfolding experiments. As a result, the values of
all force peaks from polymers of different lengths are often pooled
\includegraphics{figures/order-dep/fig}
\caption{The dependence of the unfolding force on the temporal
unfolding order for four polymers with $4$, $8$, $12$, and $16$
- molecules of identical proteins. Each point in the figure is the
+ identical protein domains. Each point in the figure is the
average of $400$ data points. The first point in each curve
represents the average of only the first peak in each of the $400$
simulated force curves, the second point represents the average of
$\kappa_\text{WLC}=203$, $207$, $161$, and $157\U{pN/nm}$,
respectively, for lengths $4$ through $16$. The insets show the
force distributions of the first, fourth, and eighth peaks, left
- to right, for the polymer with eight protein molecules. The
+ to right, for the polymer with eight protein domains. The
parameters used for generating the data were the same as those
- used for \cref{fig:sawsim:sim-sawtooth}, except the polymer
- length, and the histograms in the insets were normalized in the
- same way as in
- \cref{fig:sawsim:sim-hist}.\label{fig:sawsim:order-dep}}
+ used for \cref{fig:sawsim:sim-sawtooth}, except for the number of
+ domains. The histograms in the insets were normalized in the same
+ way as in \cref{fig:sawsim:sim-hist}.\label{fig:sawsim:order-dep}}
\end{center}
\end{figure}
increase from unfolding, and the stiffness (force constant) of the
cantilever. Among these, the effect of the cantilever force constant
is particularly interesting because cantilevers with a wide range of
-force constants are available. In addition different single molecule
+force constants are available. In addition, different single molecule
manipulation techniques, such as the AFM and laser tweezers, differ
mainly in the range of the spring constants of their force
transducers. \Cref{fig:sawsim:kappa-sawteeth} shows the simulated force
However, these simulations often involve time scales that are orders
of magnitude smaller than those of the experiments, and the parameters
used in the calculations are often neither experimentally controllable
-nor measurable. As a result, a Monte Carlo simulation approach based
-on a simple two-state kinetic model for the protein is usually used to
-analyze data from mechanical unfolding experiments. A comparison of
-the force curves measured experimentally and those generated from
-simulations can yield the unfolding rate constant of the protein in
-the absence of force as well as the distance from the native state to
-the transition state along the pulling direction. The Monte Carlo
-simulation method has been used since the first report of mechanical
-unfolding experiment using
-AFM\citep{rief97a,rief97b,rief98,carrion-vazquez99b,best02,zinober02,jollymore09},
-however, a comprehensive description and discussion of the simulation
-procedures and the intricacies involved has not been reported. In
-this paper, we provide a detailed description of the simulation
-procedure, including the theories, approximations, and assumptions
-involved. We also explain the procedure for extracting kinetic
-properties of the protein from experimental data and introduce a
-quantitative measure of fit quality between simulation and
-experimental results. In addition, the effects of various
-experimental parameters on force curve appearance are demonstrated,
-and the errors associated with different methods of data pooling are
-discussed. We believe that these results will be useful in
-experimental design, artifact identification, and data analysis for
-single molecule mechanical unfolding experiments.
+nor measurable (TODO: example parameters of each type). As a result,
+a Monte Carlo simulation approach based on a simple two-state kinetic
+model for the protein is usually used to analyze data from mechanical
+unfolding experiments. A comparison of the force curves measured
+experimentally and those generated from simulations can yield the
+unfolding rate constant of the protein in the absence of force as well
+as the distance from the native state to the transition state along
+the pulling direction. The Monte Carlo simulation method has been
+used since the first report of mechanical unfolding experiments using
+the AFM%
+\citep{rief97a,rief97b,rief98,carrion-vazquez99b,best02,zinober02,jollymore09},
+however, a comprehensive discussion of the simulation procedures and
+the intricacies involved has not been reported. In this paper, we
+provide a detailed description of the simulation procedure, including
+the theories, approximations, and assumptions involved. We also
+explain the procedure for extracting kinetic properties of the protein
+from experimental data and introduce a quantitative measure of fit
+quality between simulation and experimental results. In addition, the
+effects of various experimental parameters on force curve appearance
+are demonstrated, and the errors associated with different methods of
+data pooling are discussed. We believe that these results will be
+useful in future experimental design, artifact identification, and
+data analysis for single molecule mechanical unfolding experiments.
% simulation overview
In simulating the mechanical unfolding process, a force curve is
-generated by calculating the amount of the cantilever bending as the
+generated by calculating the amount of cantilever bending as the
substrate surface moves away from the tip. The cantilever bending is
obtained by balancing the tension in the protein polymer and the
Hookean force of the bent cantilever. The unfolding probability of
computed using any multi-dimensional root-finding algorithm.
% introduce particular models, and mention parameter aggregation
-Inside this framework, we choose a particular extension model
+Inside this framework, we chose a particular extension model
$F_i(x_i)$ for each domain state. Cantilever elasticity is described
by Hooke's law, which gives
\begin{equation}
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
+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}
% address assumptions & caveats
In the simulation, the protein polymer is assumed to be stretched in
-the direction perpendicular to the surface, which is a good
+the direction perpendicular to the substrate surface, which is a good
approximation in most experimental situations, because the unfolded
length of a protein molecule is much larger than that of the folded
form. Therefore, after one molecule unfolds, the polymer becomes much
longer and the angle between the polymer and the surface approaches
$90$ degrees\citep{carrion-vazquez00}. The joints between domain
groups are assumed to lie along a line between the surface tether
-point and the position of the tip (\cref{eq:sawsim:x-total}). The effects of
-this assumption are also minimized due to greater length of the
-unfolded domain. Finally, the interactions between different parts of
-the polymer and between the chain and the surface (except at the
-tethering points) are not considered. This is reasonable since these
-interactions should not make substantial contributions to the force
-curve at the force levels of interest, where the polymer is in a
+point and the position of the tip (\cref{eq:sawsim:x-total}). The
+effects of this assumption are also minimized due to greater length of
+the unfolded domain. Finally, the interactions between different
+parts of the polymer and between the chain and the surface (except at
+the tethering points) are not considered. This is reasonable since
+these interactions should not make substantial contributions to the
+force curve at the force levels of interest, where the polymer is in a
relatively extended conformation.
% introduce constant velocity and walk through explicit example pull
time dependence of the deflection to an exponential
function\cite{jones05}. For a $200\U{$\mu$m}$ rectangular cantilever
with a bending spring constant of $20\U{pN/nm}$, the measured
-relaxation time in water is $\sim50\U{$\mu$/s}$ (data not shown).
-This relatively large relaxation time constant makes the cantilever
-act as a low-pass filter and also causes a lag in the force
-measurement.
+relaxation time in water is $\sim50\U{$\mu$/s}$ (data not shown.
+TODO: show data). This relatively large relaxation time constant
+makes the cantilever act as a low-pass filter and also causes a lag in
+the force measurement.
\subsection{Unfolding protein molecules by force}
\label{sec:sawsim:methods-unfolding}
\cref{eq:sawsim:prob-one,eq:sawsim:prob-n} are valid, this restriction makes
time steps which should have multiple unfoldings in a single time step
highly unlikely. Experimentally measured unfolding are temporally
-supered, because the unfolding transition is characterized by
+separated, because the unfolding transition is characterized by
multiple, Markovian attempts over a large energy barrier, where the
probability of crossing the barrier in a single attempt is very low.
A successful attempt quickly extends the chain contour length,
I'm skeptical about \HTeq{8} to \HTeq{9}, so I'll rework as much of
their math as I am capable of\ldots
-\begin{align}
- \fs &= \frac{\kT}{\dx} \left[ \logp{ \frac{\r \dx}{\kexp \kT} }
- + \logp{1 + \fs\frac{\dx'}{\dx} - \frac{\FO'}{\dx} + \frac{\vD'}{\vD}\cdot\frac{\kT}{\dx}}
- + \logp{\avg{e^{\bt F_1}}}^2 \right]
- & \HTeq{8} \nonumber
-\end{align}
+\begin{multline*}
+ \fs = \frac{\kT}{\dx} \Biggl[\Biggr.
+ \logp{ \frac{\r \dx}{\kexp \kT} } \\
+ + \logp{1 + \fs\frac{\dx'}{\dx} - \frac{\FO'}{\dx} + \frac{\vD'}{\vD}\cdot\frac{\kT}{\dx}} \\
+ + \logp{\avg{e^{\bt F_1}}}^2
+ \Biggl.\Biggr]
+ \tag{\HTeq{8}}
+\end{multline*}
We simplify by dropping the 2\nd term
(``In obtaining Eq.\ \textbf{9}, we have assumed that the second term in Eq.\ \textbf{8} is small.''),
&= \frac{1}{\dx} \left( \alpha_1\rho_1 + \frac{\ep^2}{\alpha_1}
-\alpha_2\rho_2 - \frac{\ep^2}{\alpha_2} \right) \\
\ep^2\left(\frac{1}{\alpha_2} - \frac{1}{\alpha_1}\right) &= \alpha_1\rho_1 - \alpha_2\rho_2 \\
- \ep^2 \cdot \frac{\alpha_1 - \alpha_2}{\alpha_1\alpha_2} &= \\
+ \ep^2 \cdot \frac{\alpha_1 - \alpha_2}{\alpha_1\alpha_2} &= TODO\\
\ep^2 &= \frac{\alpha_1\alpha_2}{\alpha_1 - \alpha_2} \left( \alpha_1\rho_1 - \alpha_2\rho_2 \right)\\
- \ep^2 &= \frac{\kT_1\kT_2}{\kT_1 - \kT_2} \left[ \kT_1\logp{\frac{\rs1\dxs1}{\kexps1 \kT_1}}
- - \kT_2\logp{\frac{\rs2\dxs2}{\kexps2 \kT_2}} \right]
+ \begin{split}\ep^2 &= \frac{\kT_1\kT_2}{\kT_1 - \kT_2} \Biggl[\Biggr.
+ \kT_1\logp{\frac{\rs1\dxs1}{\kexps1 \kT_1}} \\
+ &\qquad- \kT_2\logp{\frac{\rs2\dxs2}{\kexps2 \kT_2}}
+ \Biggl.\Biggr]
+ \end{split}
\end{align}
Which is different from \HTeq{9} by the sign in the prefactor, and the replacement $\vD \rightarrow \kf$.
-\begin{align}
- \ep^2 &= \frac{\kT_1\kT_2}{\kT_2 - \kT_1} \left[ \kT_1\logp{\frac{\rs1\dxs1}{\vDs1 \kT_1}}
- - \kT_2\logp{\frac{\rs2\dxs2}{\vDs2 \kT_2}} \right]
- & \HTeq{9} \nonumber
-\end{align}
+\begin{equation*}
+ \ep^2 = \frac{\kT_1\kT_2}{\kT_2 - \kT_1} \left[ \kT_1\logp{\frac{\rs1\dxs1}{\vDs1 \kT_1}}
+ - \kT_2\logp{\frac{\rs2\dxs2}{\vDs2 \kT_2}} \right]
+ \tag{\HTeq{9}}
+\end{equation*}
-Alternatively, noting that \dx can vary as a function of temperature, we follow Nevo et al.\ in keeping it in.
+Alternatively, noting that \dx can vary as a function of temperature, we follow \citet{nevo05} in keeping it in.
Using $\delta \equiv \dx$
\begin{align}
0 &= \fs_1 - \fs_2 \\
&= \frac{\delta_2\alpha_1\rho_1 - \delta_1\alpha_2\rho_2}{\delta_1\delta_2} \\
\ep^2 &= \frac{\alpha_1\alpha_2}{\delta_1\alpha_1 - \delta_2\alpha_2}
\left( \delta_2\alpha_1\rho_1 - \delta_1\alpha_2\rho_2 \right)\\
- \ep^2 &= \frac{\kT_1\kT_2}{\dxs1\kT_1 - \dxs2\kT_2}
- \left[ \dxs2\kT_1\logp{\frac{\rs1\dxs1}{\kfs1 \kT_1}}
- - \dxs1\kT_2\logp{\frac{\rs2\dxs2}{\kfs2 \kT_2}} \right]
+ \begin{split}\ep^2 &= \frac{\kT_1\kT_2}{\dxs1\kT_1 - \dxs2\kT_2}
+ \Biggl[\Biggr. \dxs2\kT_1\logp{\frac{\rs1\dxs1}{\kfs1 \kT_1}} \\
+ &\qquad - \dxs1\kT_2\logp{\frac{\rs2\dxs2}{\kfs2 \kT_2}} \Biggl.\Biggr]
+ \end{split}
\end{align}
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
+but native titin contains hundreds of folded and unfolded 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
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).
+distributions (data not shown, TODO: show data).
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.
+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}
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.
+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}.
\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
+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 has been some tangential attempts towards even fancier models.
+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} introduces a smart Monte Carlo (SMC) Kramers' simulation.
+\citet{evans97} introduce a smart Monte Carlo (SMC) Kramers' simulation.
\subsection{History of experimental AFM unfolding experiments}
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.
+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}
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
+This looks similar to the Gompertz / Gumbel / Fisher-Tippett
distribution, where
\begin{align}
p(x) &\propto z\exp(-z) \\
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).
+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}
(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
+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\ldots
+yet. TODO: clean up.
\subsubsection{Cusp potentials}
projects / thesis.git / commitdiff