sawsim/discussion.tex: Standardize variables in constant loading section

Although there's too much overlap with the existing scaffold section.
The two should be combined.

diff --git a/src/sawsim/discussion.tex b/src/sawsim/discussion.tex
index 7471bae6ae46cb86a1b0f6cda5bab5b0d6ba23ca..c0a3e37cc8ce878ff150741525f905637d97155b 100644 (file)
--- a/src/sawsim/discussion.tex
+++ b/src/sawsim/discussion.tex
@@ -648,42 +648,53 @@ validate their results.
 
 \section{Single-domain proteins under constant loading}
 
-eq:sawsim:order-dep
+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, $P$ be the surviving population of folded proteins.
-Make the definitions
+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} \\
-  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}
+  \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} = kv\;,
+  \deriv{t}{F} = \deriv{x}{F}\deriv{t}{x} = \kappa v\;,
 \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 $N=1$).
+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 $N=1$).
 \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}
+    = \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}
 \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:
+Solving for theoretical histograms is merely a question of taking your
+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}
-  \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}
+  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{\p({\frac{-1}{\kappa v}\integral{}{}{F}{k_u}})} \;,
+  \label{eq:N_f}
 \end{align}
-where $c \equiv \ln(C)$ is a constant of integration scaling $P$.
+where $c \equiv \ln(C)$ is a constant of integration scaling $N_f$.
 
 \subsection{Constant unfolding rate}