Add stochastic optimization results / figure to latex document

infrared-bookchapter.tex

@@ -33,6 +33,9 @@
 \noindent\keywordname\enspace\ignorespaces#1}
 
 
+\usepackage{units}
+\newcommand{\kcalpermol}{\unit{kcal}/\unit{mol}\xspace}
+
 \usepackage{fancyvrb}
 
 \definecolor{shadecolor}{rgb}{.95, .95, .99}
@@ -647,8 +650,18 @@ Finally, we run the multi-defect optimization on our example target structures b
 multi_design_optimize(1000, 0.015)
 \end{Pythoncode}
 Note that here the number of 1000 iterations and the temperature 0.015 were chosen after some experimentation. In practice, we will often restart such procedures to obtain better solutions and/or a diverse set of good solutions. In our tests, one such optimization run took moderate run-time below 10s on current notebook hardware, while it can find very good designs for the targets. 
-The sequence \texttt{CCCCUUGCCUCAAGGGCCCUCUUCAGAGGAAGGGG} is a particularly good solution found by this strategy. For this design all three target structures have minimum free energy of -15.40 kcal/mol (at a multi-defect of 1.21).
-Even the close suboptimals contain only structures similar to the targets as can be seen from the output of \texttt{RNAsubopt} of the Vienna RNA package, which enumerates all structures within 1 kcal/mol of the minimum free energy.
+
+Figure~\ref{fig:stochasitic-optimization} shows the best multi-defects in 48 runs after up to 5120 iterations. The best sequence \texttt{CCCUGUGCUCCAUGGGCCCCCGUCAGGGGACGGGG} that was found in these 48 runs of optimization had an multi-defect of 1.31. For this sequence, the target structures have resepective energies -16.9, -16.5, and -16.9 \kcalpermol; two of the targets have minimum free energy. This small experiment yields insights into the effectivity and convergence of the optimization procedure. For applications, it seems to suggest an optimization strategy combining restarts and moderately long runs.
+
+\begin{figure}
+    \centering
+    \includegraphics[scale=0.8]{Figs/stochastic-optimization}
+    \caption{Distributions of best multi-defects from 48 runs of \texttt{multi\_design\_optimization} without stochastic optimization (step 1) and after up to 5120 optimization steps at temperature 0.015. In the box plots, the boxes extend from quartile to quartile; the medians are shownn in blue; and the whiskers reach 1.5 times beyond the quartile. Remaining data points are shown as circles.}
+    \label{fig:stochasitic-optimization}
+\end{figure}
+
+Remarkably, there are even solutions where all three targets have minimum free energy. For the sequence \texttt{CCCCUUGCCUCAAGGGCCCUCUUCAGAGGAAGGGG}, which was discovered by the same strategy, all three target structures have a free energy of -15.40 \kcalpermol (at a multi-defect of 1.21).
+Even the close suboptimals contain only structures similar to the targets as can be seen from the output of \texttt{RNAsubopt} of the Vienna RNA package, which enumerates all structures within 1 \kcalpermol of the minimum free energy.
 \begin{bashcode}
 $ RNAsubopt -s <<<CCCCUUGCCUCAAGGGCCCUCUUCAGAGGAAGGGG
 CCCCUUGCCUCAAGGGCCCUCUUCAGAGGAAGGGG -15.40   1.00