\end{align}
where $N_f$ is the number of folded domain, $\kappa$ is the spring
constant of the cantilever-polymer system, $\kappa v$ is the force
-loading rate, and $k_u$ is the unfolding rate constant
+loading rate\footnote{
+ \begin{equation}
+ \deriv{t}{F} = \deriv{x}{F}\deriv{t}{x} = \kappa v \;.
+ \end{equation}
+ Alternatively,
+ \begin{align}
+ F &= \kappa x = \kappa vt \\
+ \deriv{t}{F} &= \kappa v \;.
+ \end{align}
+ See the text before \xref{evans97}{equation}{11} or
+ \xref{dudko06}{equation}{4} for similar explanations.
+}, and $k_u$ is the unfolding rate constant
(\cref{eq:sawsim:bell}). In the last expression, $\rho\equiv
k_BT/\Delta x_u$, and $\alpha\equiv-\rho\ln(N_fk_{u0}\rho/\kappa v)$.
We can approximate $\kappa$ as a series of Hookean springs,
-\exp{\frac{F-\alpha}{\rho}}
} \;. \label{eq:sawsim:gumbel}
\end{equation}
-The distribution has a mean $\avg{F}=\alpha-\gamma_e\rho$ and a
-variance $\sigma^2 = \pi^2\rho^2/6$, where $\gamma_e=0.577\ldots$ is
-the Euler-Mascheroni constant. Therefore, the unfolding force
+The distribution has a mode $\alpha$, mean
+$\avg{F}=\alpha-\gamma_e\rho$, and a variance $\sigma^2 =
+\pi^2\rho^2/6$, where $\gamma_e=0.577\ldots$ is the Euler--Mascheroni
+constant\citep{NIST:gumbel}. Therefore, the unfolding force
distribution has a variance
\begin{equation}
\sigma^2 = \frac{\p({\frac{\pi k_BT}{\Delta x_u}})^2}{6} \;,
rate.
%
\nomenclature{$r_{uF}$}{Unfolding loading rate (newtons per second)}
-\nomenclature{$\gamma_e$}{Euler-Macheroni constant, $\gamma_e=0.577\ldots$}
+\nomenclature{$\alpha$}{The mode unfolding force,
+ $\alpha\equiv-\rho\ln(N_f k_{u0}\rho/\kappa v)$
+ (\cref{eq:sawsim:gumbel}).}
+\nomenclature{$\rho$}{The characteristic unfolding force,
+ $\rho\equiv k_BT/\Delta x_u$ (\cref{eq:sawsim:gumbel}).}
+\nomenclature{$\gamma_e$}{Euler--Macheroni constant, $\gamma_e=0.577\ldots$}
\nomenclature{$\sigma$}{Standard deviation. For example, $\sigma$ is
used as the standard deviation of an unfolding force distribution in
\cref{eq:sawsim:gumbel}. Not to be confused with the photodiode
force histograms with analytical histograms for a number of situations
where solving for the analytical histogram is possible.
-\section{Review of current research}
-
-There are two main approaches to modeling protein domain unfolding
-under tension: Bell's and Kramers'\citep{schlierf06,hummer03,dudko06}.
-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{rief97a,carrion-vazquez99b,schlierf06} due to its
-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.
-
-\subsection{Evolution of unfolding modeling}
-
-Evans introduced the saddle-point Kramers' approximation in a protein unfolding context in 1997 (\citet{evans97} Eqn.~3).
-However, early work on mechanical unfolding focused on the simpler Bell model\citep{rief97a}.%TODO
-In the early 2000's, the saddle-point/steepest-descent approximation to Kramer's model (\xref{hanggi90}{equation}{4.56c}) was introduced into our field\citep{dudko03,hyeon03}.%TODO
-By the mid 2000's, the full-blown double-integral form of Kramer's model (\xref{hanggi90}{equation}{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}
-
-There is a long history of protein unfolding and unbinding
-simulations. Early work by \citet{grubmuller96} and
-\citet{izrailev97} focused on molecular dynamics (MD) simulations of
-receptor-ligand breakage. Around the same time, \citet{evans97}
-introduced a Monte Carlo Kramers' simulation in the context of
-receptor-ligand breakage. The approach pioneered by \citet{evans97}
-was used as the basis for analysis of the initial protein unfolding
-experiments\citep{rief97a}. However, none of these earlier
-implementations were open source, which made it difficult to reuse or
-validate their results.
-%
-\nomenclature{MD}{Molecular dynamics simulation. Simulate the
- physical motion of atoms and molecules by numerically solving
- Newton's equations.}
-
-\subsection{History of experimental AFM unfolding experiments}
-
-\begin{itemize}
- \item \citet{rief97a}:
-\end{itemize}
-
-\subsection{History of experimental laser tweezer unfolding experiments}
-
-\begin{itemize}
- \item \citet{izrailev97}:
-\end{itemize}
-
-\section{Single-domain proteins under constant loading}
-
-TODO: consolidate with \cref{sec:sawsim:results:scaffold}.
-
-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, $N_f$ be
-the surviving population of folded proteins. Make the definitions
-\begin{align}
- v &\equiv \deriv{t}{x} && \text{the pulling velocity} \\
- \kappa &\equiv \deriv{x}{F} && \text{the loading spring constant} \\
- N_{f0} &\equiv N_f(t=0) && \text{the initial number of folded proteins} \\
- N_u &\equiv N_{f0} - N_f && \text{the number of unfolded proteins} \\
- k_u &\equiv -\frac{1}{N_f} \deriv{t}{N_f} && \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} = \kappa v\;,
-\end{equation}
-a constant, since both $\kappa$ and $v$ are constant (\citet{evans97}
-in the text on the first page, \citet{dudko06} in the text just before
-\fref{equation}{4}).
-
The instantaneous likelyhood of a protein unfolding is given by
$\deriv{F}{N_u}$, and the unfolding histogram is merely this function
-discretized over a bin of width $W$ (This is similar to
-\xref{dudko06}{equation}{2}, remembering that $\dot{F}=\kappa v$, that
-their probability density is not a histogram ($W=1$), and that their
-probability density function is normalized to $N=1$).
+discretized over a bin of width $W$\footnote{
+ This is similar to \xref{dudko06}{equation}{2}, remembering that
+ $\dot{F}=\kappa v$, that their probability density is not a
+ histogram ($W=1$), and that their probability density function is
+ normalized to $N=1$
+}.
\begin{equation}
h(F) \equiv \deriv{\text{bin}}{F}
= \deriv{F}{N_u} \cdot \deriv{\text{bin}}{F}
= W \deriv{F}{N_u}
= -W \deriv{F}{N_f}
= -W \deriv{t}{N_f} \deriv{F}{t}
- = \frac{W}{vk} N_f\kappa \label{eq:unfold:hist}
+ = \frac{W}{\kappa v} N_f k_u \label{eq:unfold:hist}
\end{equation}
Solving for theoretical histograms is merely a question of taking your
-chosen $k_u$, solving for $N_f(f)$, and plugging into
+chosen $k_u$, solving for $N_f(F)$, and plugging into
\cref{eq:unfold:hist}. We can also make a bit of progress solving for
$N_f$ in terms of $k_u$ as follows:
\begin{align}
k_u &\equiv -\frac{1}{N_f} \deriv{t}{N_f} \\
-k_u \dd t \cdot \deriv{t}{F} &= \frac{\dd N_f}{N_f} \\
- \frac{-1}{\kappa v} \int k_0 \dd F &= \ln(N_f) + c \\
- N_f &= C\exp{\frac{-1}{\kappa v}\integral{}{}{F}{k_u}} \;,
+ \frac{-1}{\kappa v} \integral{0}{F}{F'}{k_0(F')}
+ &= \left. \ln(N_f(F')) \right|_0^F
+ = \ln\p({\frac{N_f(F)}{N_f(0)}})
+ = \ln\p({\frac{N_f(F)}{N}}) \\
+ N_f(F) &= N\exp{\frac{-1}{\kappa v}\integral{0}{F}{F'}{k_u(F')}} \;,
\label{eq:N_f}
\end{align}
-where $c \equiv \ln(C)$ is a constant of integration scaling $N_f$.
+where $N_f(0) = N$ because all the domains are initially folded.
+%
+\nomenclature{$W$}{Bin width of an unfolding force histogram
+ (\cref{eq:unfold:hist}).}
-\subsection{Constant unfolding rate}
+\subsubsection{Constant unfolding rate}
-In the extremely weak tension regime, the proteins' unfolding rate is independent of tension, we have
+In the extremely weak tension regime, the proteins' unfolding rate is
+independent of tension, so we can simplify \cref{eq:N_f} and plug into
+\cref{eq:unfold:hist}.
\begin{align}
- P &= C\exp{\frac{-1}{kv}\integral{}{}{F}{\kappa}}
- = C\exp{\frac{-1}{kv}\kappa F}
- = C\exp{\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{\frac{-\kappa F}{kv}}
+ N_f &= N\exp{\frac{-1}{\kappa v}\integral{0}{F}{F'}{\colA{k_u(F')}}}
+ = N\exp{\frac{-\colA{k_{u0}}}{\kappa v}\colB{\integral{0}{F}{F'}{}}}
+ = N\exp{\frac{-k_{u0} \colB{F}}{\kappa v}} \\
+ h(F) &= \frac{W}{\kappa v} N_f k_u
+ = \frac{W k_{u0} N}{\kappa v} \exp{\frac{-k_{u0} F}{\kappa v}} \;.
\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.
+A constant unfolding-rate/hazard-function gives exponential decay.
+This is not an 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}
+\subsubsection{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}).
+(\cref{sec:sawsim:rate:bell}). We can simplify the following
+calculation by parametrizing with the characteristic force $\rho$
+defined in \cref{sec:sawsim:results:scaffold} and the similar
+single-domain mode $\alpha'\equiv-\rho\ln(k_{u0}\rho/\kappa v)$. With
+these substitutions, \cref{eq:sawsim:bell} becomes
\begin{equation}
- \kappa = \kappa_0 \cdot \exp{\frac{F \dd x}{k_B T}}
- = \kappa_0 \cdot \exp{a F} \;,
+ k_u = k_{u0} \exp{\frac{F}{\rho}} \;.
\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
+The unfolding histogram is then given via \cref{eq:N_f,eq:unfold:hist}.
\begin{align}
- P &= C\exp{\frac{-1}{kv}\integral{}{}{F}{\kappa}}
- = C\exp{\frac{-1}{kv} \frac{\kappa_0}{a} \exp{a F}}
- = C\exp{\frac{-\kappa_0}{akv}\exp{a F}} \\
- P(0) &\equiv P_0 = C\exp{\frac{-\kappa_0}{akv}} \\
- C &= P_0 \exp{\frac{\kappa_0}{akv}} \\
- P &= P_0 \exp{\frac{\kappa_0}{akv}\p({1-\exp{a F}})} \\
- h(F) &= \frac{W}{vk} P \kappa
- = \frac{W}{vk} P_0
- \exp{\frac{\kappa_0}{akv}\p({1-\exp{a F}})} \kappa_0 \exp{a F}
- = \frac{W\kappa_0 P_0}{vk}
- \exp{a F + \frac{\kappa_0}{akv}\p({1-\exp{a F}})} \;.
+ N_f &= N\exp{\frac{-1}{\kappa v}\integral{0}{F}{F'}{\colA{k_u}}}
+ = N\exp{\frac{-1}{\kappa v}
+ \integral{0}{F}{F'}{\colAB{k_{u0}}{\colA{\exp{\frac{F'}{\rho}}}}}}
+ = N\exp{\frac{-\colB{k_{u0}}}{\kappa v}
+ \colA{\integral{0}{F}{F'}{\exp{\frac{F'}{\rho}}}}}
+ = N\exp{\frac{\colB{-}k_{u0}\colA{\rho}}{\kappa v}
+ \colAB{\p({\exp{\frac{F}{\rho}}-1})}} \\
+ &= N\exp{\colA{\frac{k_{u0}\rho}{\kappa v}}
+ \colB{\p({1 - {\exp{\frac{F}{\rho}}}})}}
+ = N\exp{\colAB{\exp{\frac{-\alpha'}{\rho}}}
+ \colB{\p({1 - {\exp{\frac{F}{\rho}}}})}}
+ = N\exp{\colB{\exp{\frac{-\alpha'}{\rho}} -
+ \exp{\frac{F-\alpha'}{\rho}}}} \\
+ h(F) &= \frac{W}{\kappa v} \colA{N_f} \colB{k_u}
+ = \frac{W}{\kappa v}
+ \colA{N\exp{\exp{\frac{-\alpha'}{\rho}} - \exp{\frac{F-\alpha'}{\rho}}}}
+ \colB{k_{u0}\exp{\frac{F}{\rho}}}
+ = \frac{W N \colAB{k_{u0}}}{\colA{\kappa v}}
+ \exp{\colB{\frac{F}{\rho}} - \exp{\frac{F-\alpha'}{\rho}} +
+ \exp{\frac{-\alpha'}{\rho}}} \\
+ &= \frac{W N}{\colA{\rho}}
+ \exp{\frac{F \colA{-\alpha'}}{\rho} - \exp{\frac{F-\alpha'}{\rho}} +
+ \colB{\exp{\frac{-\alpha'}{\rho}}}}
+ = \frac{W N}{\rho}
+ \exp{\frac{F-\alpha'}{\rho} - \exp{\frac{F-\alpha'}{\rho}}}
+ \colB{\exp{\exp{\frac{-\alpha'}{\rho}}}} \\
+ &= \frac{W N \exp{\exp{\frac{-\alpha'}{\rho}}}}{\rho}
+ \exp{\frac{F-\alpha'}{\rho} - \exp{\frac{F-\alpha'}{\rho}}}
\label{eq:unfold:bell_pdf}
\end{align}
-The $F$ dependent behavior reduces to
-\begin{equation}
- h(F) \propto \exp{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{-\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{\frac{x-\alpha}{\beta}
- -\exp{\frac{x-\alpha}{\beta}}}
- \label{eq:sawsim:gumbel-x}
-\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
-\nomenclature{$\mu$}{The mean of a distribution (e.g. the Gumbel
- distribution, \cref{eq:sawsim:gumbel-x}).}
-\begin{align}
- P(F)
- &= \frac{1}{\beta} \exp{\frac{F+\beta\ln(\kappa\beta/kv)}{\beta}
- -\exp{\frac{F+\beta\ln(\kappa\beta/kv)}{\beta}}} \\
- &= \frac{1}{\beta} \exp{F/\beta}\exp{\ln(\kappa\beta/kv)}
- \exp{-\exp{F/\beta}\exp{\ln(\kappa\beta/kv)}} \\
- &= \frac{1}{\beta} \frac{\kappa\beta}{kv} \exp{F/\beta}
- \exp{-\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.
-
-% Consolidate with src/sawsim/discussion.tex
+which matches \cref{eq:sawsim:gumbel} except for a constant
+prefactor due to the range\footnote{
+ The Gumbel distribution in \cref{eq:sawsim:gumbel} is normalized for
+ the range $-\infty < F < \infty$, but \cref{eq:unfold:bell_pdf} is
+ normalized for the range $0 \le F < \infty$.
+}.
+%
+\nomenclature{$\alpha'$}{The mode unfolding force for a single folded
+ domain, $\alpha'\equiv-\rho\ln(k_{u0}\rho/\kappa v)$
+ (\cref{eq:unfold:bell_pdf}).}
-\subsection{Saddle-point Kramers' model}
+\subsubsection{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).
\nomenclature{$l_{ts}$}{The characteristic length of the transition
state (\cref{eq:kramers-saddle}).}
-\citet{evans97} solved this unfolding rate for both inverse power law potentials and cusp potentials.
+\citet{evans97} solved this unfolding rate for both inverse power law
+potentials and cusp potentials.
-\section{Double-integral Kramers' theory}
+\subsubsection{Double-integral Kramers' theory}
The double-integral form of overdamped Kramers' theory may be too
complex for analytical predictions of unfolding-force histograms.
Looking for analytic solutions to Kramers' $k(F)$, we find that there
are not many available in a closed form. However, we do have analytic
solutions for unforced $k$ for cusp-like and quartic potentials.
+
+\section{Review of current research}
+
+There are two main approaches to modeling protein domain unfolding
+under tension: Bell's and Kramers'\citep{schlierf06,hummer03,dudko06}.
+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{rief97a,carrion-vazquez99b,schlierf06} due to its
+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.
+
+\subsection{Evolution of unfolding modeling}
+
+Evans introduced the saddle-point Kramers' approximation in a protein unfolding context in 1997 (\citet{evans97} Eqn.~3).
+However, early work on mechanical unfolding focused on the simpler Bell model\citep{rief97a}.%TODO
+In the early 2000's, the saddle-point/steepest-descent approximation to Kramer's model (\xref{hanggi90}{equation}{4.56c}) was introduced into our field\citep{dudko03,hyeon03}.%TODO
+By the mid 2000's, the full-blown double-integral form of Kramer's model (\xref{hanggi90}{equation}{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}
+
+There is a long history of protein unfolding and unbinding
+simulations. Early work by \citet{grubmuller96} and
+\citet{izrailev97} focused on molecular dynamics (MD) simulations of
+receptor-ligand breakage. Around the same time, \citet{evans97}
+introduced a Monte Carlo Kramers' simulation in the context of
+receptor-ligand breakage. The approach pioneered by \citet{evans97}
+was used as the basis for analysis of the initial protein unfolding
+experiments\citep{rief97a}. However, none of these earlier
+implementations were open source, which made it difficult to reuse or
+validate their results.
+%
+\nomenclature{MD}{Molecular dynamics simulation. Simulate the
+ physical motion of atoms and molecules by numerically solving
+ Newton's equations.}
+
+\subsection{History of experimental AFM unfolding experiments}
+
+\begin{itemize}
+ \item \citet{rief97a}:
+\end{itemize}
+
+\subsection{History of experimental laser tweezer unfolding experiments}
+
+\begin{itemize}
+ \item \citet{izrailev97}:
+\end{itemize}
projects / thesis.git / commitdiff