sawsim/discussion.tex: Talk about benedetti11 in relation to my analysis

diff --git a/src/sawsim/discussion.tex b/src/sawsim/discussion.tex
index 832c6bb50f6f9ef984202a728a9ddc86a7431901..7c3747db6d2d78ee37807ae9f079ba4225a5e880 100644 (file)
--- a/src/sawsim/discussion.tex
+++ b/src/sawsim/discussion.tex
@@ -155,14 +155,20 @@ The rate of unfolding events with respect to force is
     &= \frac{N_f k_{u0}}{\kappa v}\exp\p({\frac{F\Delta x_u}{k_B T}})
     = \frac{1}{\rho}\exp\p({\frac{F-\alpha}{\rho}}) \;,
 \end{align}
-where $N_f$ is the number of folded domain,
-$\kappa=(1/\kappa_c+N_u/\kappa_\text{WLC})^{-1}$ is the spring
-constant of the cantilever-polymer system ($\kappa_\text{WLC}$ is the
-effective spring constant of one unfolded domain, assumed constant for
-a particular polymer/cantilever combination), $\kappa v$ is the force
+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
 (\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,
+\begin{equation}
+  \kappa=\p({\frac{1}{\kappa_c}+\frac{N_u}{\kappa_\text{WLC}}})^{-1} \;,
+  \label{eq:kappa-system}
+\end{equation}
+where $\kappa_\text{WLC}$ is the effective spring constant of one
+unfolded domain, assumed constant for a particular polymer/cantilever
+combination.
+
 The event probability density for events with an exponentially
 increasing likelihood function follows the Gumbel (minimum)
 probability density\citep{NIST:gumbel} with $\rho$ and $\alpha$ being
@@ -196,6 +202,7 @@ constant, as we have assumed here, but a non-linear function of $F$.
 as far as I know, nobody has found an analytical form for the
 unfolding force histograms produced under such a variable loading
 rate.
+%
 \nomenclature{$r_{uF}$}{Unfolding loading rate (newtons per second)}
 \nomenclature{$\gamma_e$}{Euler-Macheroni constant, $\gamma_e=0.577\ldots$}
 
@@ -253,6 +260,43 @@ length.
   \end{center}
 \end{figure}
 
+\citet{benedetti11} have since proposed an alternative
+parameterization for \cref{eq:kappa-system}, using
+\begin{equation}
+  \kappa = \p({\frac{1}{\kappa_c}
+               + \frac{N_f}{\kappa_f} + \frac{N_u}{\kappa_u}})^{-1}
+    \equiv \frac{\kappa'}{1 - A N_f} \;,
+  \label{eq:kappa-system-benedetti}
+\end{equation}
+where $\kappa'$ is the spring constant of the completely unfolded
+chain and $A$ is a correction term for the supramolecular scaffold.
+This is effectively a first order Taylor expansion for $\kappa^{-1}$
+about $N_f=0$, but the remaining analysis is identical.
+\begin{align}
+  f(N_f) \equiv \kappa^{-1}
+    &= \frac{1}{\kappa_c} + \frac{N_f}{\kappa_f} + \frac{N - N_f}{\kappa_u} \\
+    &= f(0) + \left.\deriv{N_f}{f}\right|_{N_f=0} N_f + \order{N_f^2} \\
+    &\approx \p({\frac{1}{\kappa_c} + \frac{N}{\kappa_u}}) +
+             \p({\frac{1}{\kappa_f} - \frac{1}{\kappa_u}}) N_f
+  \label{eq:kappa-system-taylor}
+\end{align}
+In the case where the wormlike chain stiffnesses $\kappa_f$ and
+$\kappa_u$ are fairly constant over the unfolding region, there are no
+higher order terms and the first order expansion in
+\cref{eq:kappa-system-taylor} is exact.  Comparing
+\cref{eq:kappa-system-benedetti,eq:kappa-system-taylor}, we see
+\begin{align}
+  \kappa' &= \frac{1}{\kappa_c} + \frac{N}{\kappa_u} \\
+  -\kappa' A &= \frac{1}{\kappa_f} - \frac{1}{\kappa_u} \\
+  A &= \frac{\frac{1}{\kappa_u} - \frac{1}{\kappa_f}}
+            {\frac{1}{\kappa_c} + \frac{N}{\kappa_u}}
+\end{align}
+By focusing on the $A=0$ case (\ie~$\kappa_f=\kappa_u$),
+\citet{benedetti11} avoid running Monte Carlo simulations when
+modeling unfolding histograms.  This simplification does not hold for
+our simulated data (\cref{fig:sawsim:order-dep}), but for some
+experimental analysis the loss of accuracy may be acceptable in return
+for the reduced computational cost.
 
 \subsection{The effect of cantilever force constant}
 \label{sec:sawsim:cantilever}