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For more info see http://www.lyx.org/ \lyxformat 245 \begin_document \begin_header \textclass revtex4 \begin_preamble % PRL look and style (easy on the eyes) % Two-column style (for submission/review/editing) %\documentclass[aps,prl,preprint,nofootinbib,superscriptaddress]{revtex4} \usepackage{palatino} %\usepackage[mathbf,mathcal]{euler} %\usepackage{citesort} \usepackage{dcolumn} \usepackage{boxedminipage} \usepackage[colorlinks=true,citecolor=blue,linkcolor=blue]{hyperref} \bibliographystyle{apsrevlong} % italicized boldface for math (e.g. vectors) \newcommand{\bfv}[1]{{\mbox{\boldmath{$#1$}}}} % non-italicized boldface for math (e.g. matrices) \newcommand{\bfm}[1]{{\bf #1}} %\newcommand{\bfm}[1]{{\mbox{\boldmath{$#1$}}}} %\newcommand{\bfm}[1]{{\bf #1}} \newcommand{\expect}[1]{\left \langle #1 \right \rangle} % <.> for denoting expectations over realizations of an experiment or thermal averages % vectors %\newcommand{\x}{\bfv{x}} %\newcommand{\y}{\bfv{y}} %\newcommand{\f}{\bfv{f}} %\newcommand{\z}{\bfv{z}} %\newcommand{\p}{\bfv{p}} %\newcommand{\q}{\bfv{q}} %\newcommand{\T}{\mathrm{T}} \newcommand{\op}[1]{\mathcal{#1}} %\newcommand{\vektor}[1]{{\bf #1}} %\renewcommand{\vec}[1]{\bfm{#1}} % italicized boldface %\renewcommand{\vec}[1]{{\bf #1}} \renewcommand{\vec}[1]{{\mbox{\boldmath{$#1$}}}} \newcommand{\q}{\vec{q}} % coordinates \newcommand{\p}{\vec{p}} % momenta \newcommand{\z}{\vec{z}} % phase space point \newcommand{\x}{\vec{x}} % general vector x \newcommand{\M}{\bfm{M}} % diagonal mass matrix \newcommand{\grad}{\nabla} \newcommand{\timeavg}[1]{\overline{#1}} % time average over a trajectory \newcommand{\bfc}{\bfm{c}} \newcommand{\hatf}{\hat{f}} \newcommand{\bTheta}{\bfm{\Theta}} \newcommand{\btheta}{\bfm{\theta}} \newcommand{\bhatf}{\bfm{\hat{f}}} \newcommand{\Cov}[1]{\mathrm{cov}\left( #1 \right)} \newcommand{\Ept}[1]{{\mathrm E}\left[ #1 \right]} \newcommand{\Eptk}[2]{{\mathrm E}\left[ #2 \,|\, #1\right]} \newcommand{\T}{\mathrm{T}} % T used in matrix transpose %% DOCUMENT %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \end_preamble \options aps,pre,nofootinbib,superscriptaddress \language english \inputencoding auto \fontscheme default \graphics default \paperfontsize default \spacing single \papersize default \use_geometry false \use_amsmath 1 \cite_engine natbib_authoryear \use_bibtopic false \paperorientation portrait \secnumdepth 3 \tocdepth 3 \paragraph_separation indent \defskip medskip \quotes_language english \papercolumns 2 \papersides 1 \paperpagestyle default \tracking_changes false \output_changes false \author "" \author "" \end_header \begin_body \begin_layout Standard \begin_inset ERT status inlined \begin_layout Standard %% TITLE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \end_layout \end_inset \end_layout \begin_layout Title Optimal use of data in parallel tempering simulations for the construction of kinetic models of peptide dynamics \end_layout \begin_layout Author Jan-Hendrik Prinz \end_layout \begin_layout Author Email jan-hendrik.prinz@iwr.uni-heidelberg.de \end_layout \begin_layout Affiliation IWR, University of Heidelberg, INF 368, 69120 Heidelberg, Germany \end_layout \begin_layout Author John D. Chodera \end_layout \begin_layout Thanks Corresponding author \end_layout \begin_layout Author Email jchodera@stanford.edu \end_layout \begin_layout Affiliation Department of Chemistry, Stanford University, Stanford, CA 94305 \end_layout \begin_layout Author Vijay S. Pande \end_layout \begin_layout Author Email pande@stanford.edu \end_layout \begin_layout Affiliation Department of Chemistry, Stanford University, Stanford, CA 94305 \end_layout \begin_layout Author Frank Noé \end_layout \begin_layout Author Email noe@math.fu-berlin.de \end_layout \begin_layout Affiliation DFG Research Center Matheon, FU Berlin, Arnimallee 6, 14195 Berlin, Germany \end_layout \begin_layout Date \begin_inset ERT status collapsed \begin_layout Standard \backslash today \end_layout \end_inset \end_layout \begin_layout Abstract \begin_inset ERT status inlined \begin_layout Standard %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% \end_layout \begin_layout Standard % ABSTRACT/pacs \end_layout \begin_layout Standard %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% \end_layout \end_inset \end_layout \begin_layout Abstract - Recently, others have demonstrated how the short physical trajectories generated in parallel tempering simulations between exchanges can be used to construct Markov models to describe peptide dynamics at each simulated temperature. \end_layout \begin_layout Abstract - While able to describe the temperature-dependent kinetics, this approach does not make optimal use of all available data, instead parameterizing the model at a given temperature with only data from that temperature. \end_layout \begin_layout Abstract - We propose a strategy where, with a simple modification of the parallel tempering protocol, the harvested trajectories can be reweighted, allowing data from all temperatures to contribute to the estimated kinetic model without requiring the assumption of a particular kinetic model (like Arrhenius) beyond that of classical statistical mechanics. \end_layout \begin_layout Abstract - This method not only reduces the statistical uncertainty in the kinetic model, providing estimates of transition rates even for transitions not observed at the temperature of interest, it allows the kinetic model to be estimated at temperatures other than those simulated from. \end_layout \begin_layout Abstract - To demonstrate the method in a system, we apply the method to the solvated terminally-blocked alanine peptide. \end_layout \begin_layout Standard \begin_inset ERT status inlined \begin_layout Standard % \backslash pacs{02.30.Cj,05.70.Ce,82.20.Wt} \end_layout \end_inset \begin_inset ERT status inlined \begin_layout Standard %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% \end_layout \begin_layout Standard % INTRODUCTION \end_layout \begin_layout Standard %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% \end_layout \end_inset \end_layout \begin_layout Section Introduction \end_layout \begin_layout Standard \begin_inset LatexCommand \label{section:introduction} \end_inset \end_layout \begin_layout Itemize Markov models provide a way to summarize the conformational dynamics of biomolecules such as peptides and proteins. \end_layout \begin_layout Itemize Recently, an approach based on using short physical trajectories generated from parallel-tempering simulations has been proposed that allows both thermodynamics and kinetics to be extracted, but does not make optimal use of all the data. \end_layout \begin_layout Itemize We propose an approach to use data from all temperatures, with a slight modification of the protocol, to construct an estimate of the Markov model for any temperature desired, without using assumptions about the kinetics \end_layout \begin_layout Itemize Outline of this manuscript \end_layout \begin_layout Standard \begin_inset ERT status inlined \begin_layout Standard %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% \end_layout \begin_layout Standard % THEORY \end_layout \begin_layout Standard %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% \end_layout \end_inset \end_layout \begin_layout Section Theory \end_layout \begin_layout Standard \begin_inset LatexCommand \label{section:theory} \end_inset \end_layout \begin_layout Subsection Discrete-state Markovian kinetic models \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard { \end_layout \end_inset \begin_inset ERT status collapsed \begin_layout Standard \backslash color{red} \end_layout \end_inset Somewhere in this section, we show how the (row-stochastic) transition probability can be defined as \begin_inset Formula \begin{eqnarray} T_{ij}(\tau;\beta) & = & \frac{C_{ij}(\tau;\beta)}{\sum\limits _{k}C_{ik}(\tau;\beta)}\label{equation:transition-matrix-definition}\end{eqnarray} \end_inset where the state-to-state time-correlation function \begin_inset Formula $C_{ij}(\tau;\beta)$ \end_inset is given by \begin_inset Formula \begin{eqnarray} C_{ij}(\tau;\beta) & \equiv & \expect{\chi_{i}(0)\,\chi_{j}(\tau)}_{\beta}\end{eqnarray} \end_inset and is symmetric --- \begin_inset Formula $C_{ij}(\tau;\beta)=C_{ji}(\tau;\beta)$ \end_inset --- for systems obeying detailed balance. \begin_inset ERT status collapsed \begin_layout Standard } \end_layout \end_inset \end_layout \begin_layout Subsection Transition probabilities by reweighting \end_layout \begin_layout Subsubsection Dynamical reweighting \end_layout \begin_layout Standard Suppose we have a set of \begin_inset Formula $N_{k}$ \end_inset Hamiltonian trajectories \begin_inset Formula $\z_{kn}(t)$ \end_inset , \begin_inset Formula $n=1,\ldots,N_{k}$ \end_inset , \begin_inset Formula $t\in[0,T]$ \end_inset , where the initial phase space points \begin_inset Formula $\z_{kn}(0)$ \end_inset are sampled from canonical (NVT) distributions at corresponding inverse temperatures \begin_inset Formula $\beta_{k}$ \end_inset , \begin_inset Formula $k=1,\ldots,K$ \end_inset . By the application of \emph on dynamical reweighting \emph default \InsetSpace ~ \begin_inset LatexCommand \cite{shirts-chodera:jcp:2008:mbar} \end_inset , we can estimate a correlation function \begin_inset Formula $C_{AB}(\tau;\beta)$ \end_inset using this set of trajectories as \begin_inset Formula \begin{eqnarray} C_{AB}(\tau;\beta) & \equiv & [Z(\beta)]^{-1}\int d\z_{0}\, e^{-\beta H(\z_{0})}\,\mathcal{A}[\z(t)]\nonumber \\ & \approx & \sum\limits _{k=1}^{K}\sum\limits _{n=1}^{N_{k}}w_{kn}(\beta)\,\mathcal{A}[\z_{kn}(t)]\end{eqnarray} \end_inset where \begin_inset Formula $\mathcal{A}[\z(t)]$ \end_inset is a functional of the trajectory \begin_inset Formula $\z(t)$ \end_inset whose expectation produces the desired correlation function, such as \begin_inset Formula $\mathcal{A}[\z(t)]=A(\z(0))\, B^{*}(\z(\tau))$ \end_inset . The normalized trajectory weights \begin_inset Formula $w_{kn}(\beta)$ \end_inset are specified in terms of unnormalized weights \begin_inset Formula $\tilde{w}_{kn}(\beta)$ \end_inset \begin_inset Formula \begin{eqnarray} \tilde{w}_{kn}(\beta) & = & \left[\sum_{k'=1}^{K}N_{k'}\,\exp[f_{k'}-(\beta-\beta_{k'})\, E_{kn}]\right]^{-1}\end{eqnarray} \end_inset such that \begin_inset Formula $w_{kn}(\beta)=\tilde{w}_{kn}(\beta)/\sum\limits _{k=1}^{K}\sum\limits _{n=1}^{N_{k}}\tilde{w}_{kn}(\beta)$ \end_inset , with \begin_inset Formula $E_{kn}\equiv H(\z_{kn}(0))$ \end_inset denoting the total energy of the trajectory. \end_layout \begin_layout Standard The dimensionless free energies \begin_inset Formula $f_{i}=-\ln Z_{i}+c$ \end_inset are determined by solution of a set of self-consistent equations \begin_inset Formula \begin{eqnarray} e^{-f_{i}} & = & \sum_{k=1}^{K}\sum_{n=1}^{N_{k}}\tilde{w}_{kn}(\beta)\end{eqnarray} \end_inset which can be done efficiently by a number of means\InsetSpace ~ \begin_inset LatexCommand \cite{shirts-chodera:jcp:2008:mbar} \end_inset . \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard %The statistical uncertainty in the estimate of $C_{AB}( \backslash tau; \backslash beta)$ can also be estimated by forming the matrix $ \backslash bfm{ \backslash Theta}$: \end_layout \begin_layout Standard % \backslash begin{eqnarray} \end_layout \begin_layout Standard % \backslash Theta_{ij} &=& \end_layout \begin_layout Standard % \backslash end{eqnarray} \end_layout \end_inset \end_layout \begin_layout Subsubsection Modified parallel tempering protocol \end_layout \begin_layout Standard By using a modified parallel tempering protocol, we can generate a set of Hamiltonian trajectory segments \begin_inset Formula $\z_{kn}(t)$ \end_inset of uniform length \begin_inset Formula $T\ge\tau$ \end_inset whose initial phase space points \begin_inset Formula $\z_{kn}(0)$ \end_inset are distributed from the canonical (NVT) ensemble at corresponding temperatures \begin_inset Formula $\beta_{1},\ldots,\beta_{K}$ \end_inset . \end_layout \begin_layout Standard We start assuming that some process was used to generate initial phase space points \begin_inset Formula $\z_{k0}(0)$ \end_inset from equilibrium at each corresponding inverse temperature \begin_inset Formula $\beta_{k}$ \end_inset \begin_inset Formula \begin{eqnarray} \z_{k0}(0) & \sim & [Z(\beta_{k})]^{-1}\, e^{-\beta H(\z_{k0}(0))}\end{eqnarray} \end_inset This could be done by means of a standard parallel tempering protocol, or by running the modified protocol for a number of iterations starting from a single configuration. \end_layout \begin_layout Standard Consider iteration \begin_inset Formula $n$ \end_inset of the algorithm. For each temperature index \begin_inset Formula $k=1,\ldots,K$ \end_inset , we propagate Hamilton's equations of motion using a symplectic integrator to generate trajectories of \begin_inset Formula $\z_{kn}(t)$ \end_inset of length \begin_inset Formula $T$ \end_inset . Finally, we propose exchanges between the final configurations \begin_inset Formula $\z_{in}(\tau)$ \end_inset and \begin_inset Formula $\z_{jn}(\tau)$ \end_inset of temperatures \begin_inset Formula $\beta_{i}$ \end_inset and \begin_inset Formula $\beta_{j}$ \end_inset , accepting or rejecting the exchange with the Metropolis-like probability depending on the final potential energies of the configurations \begin_inset Formula $U_{i}$ \end_inset and \begin_inset Formula $U_{j}$ \end_inset with \begin_inset Formula \begin{eqnarray*} P_{\mathrm{exch}}(U_{i},\beta_{i};U_{j},\beta_{j}) & = & \min\left\{ 1,\exp[-(\beta_{i}-\beta_{j})(U_{j}-U_{i})]\right\} \end{eqnarray*} \end_inset Regardless of whether the exchange is accepted or rejected, we reassign the velocities according to the Maxwell-Boltzmann distribution at the new (old) temperatures, and denote the new phase space points as \begin_inset Formula $\z_{k(n+1)}(0)$ \end_inset , from which the next iteration can be carried out. \end_layout \begin_layout Subsubsection Transition probabilities \end_layout \begin_layout Standard To estimate the transition probabilities \begin_inset Formula $T_{ij}(\tau;\beta)$ \end_inset , we first compute an estimate of the correlation functions \begin_inset Formula $C_{ij}(\tau;\beta)$ \end_inset \begin_inset Formula \begin{eqnarray} \hat{C}_{ij}(\tau;\beta) & = & \sum_{k=1}^{K}\sum_{n=1}^{N_{k}}w_{kn}(\beta)\, A_{kn}(\tau)\end{eqnarray} \end_inset where we have defined the per-trajectory observable \begin_inset Formula $A_{kn}(\tau)$ \end_inset to make use of overlapping trajectory segments (in the case \begin_inset Formula $T>\tau$ \end_inset ) and the time-reversibility of Hamiltonian dynamics, thus ensuring \begin_inset Formula $\hat{C}_{ij}(\tau;\beta)=\hat{C}_{ji}(\tau;\beta)$ \end_inset : \begin_inset Formula \begin{eqnarray} A_{kn}(t) & \equiv & \frac{1}{2(T-t)}\int_{0}^{\tau-t}dt_{0}\,[\chi_{i}(\z_{kn}(t_{0}))\,\chi_{j}(\z_{kn}(t_{0}+t))\nonumber \\ & & \mbox{}+\chi_{j}(\z_{kn}(t_{0}))\chi_{i}(\z_{kn}(t_{0}+t))]\end{eqnarray} \end_inset The row-stochastic transition matrix estimate \begin_inset Formula $\hat{\bfm{T}}(\tau;\beta)$ \end_inset is then estimated from Eq.\InsetSpace ~ \begin_inset LatexCommand \ref{equation:transition-matrix-definition} \end_inset \begin_inset Formula \begin{eqnarray} \hat{T}_{ji}(\tau;\beta) & = & \frac{\hat{C}_{ij}(\tau;\beta)}{\sum\limits _{k}\hat{C}_{ik}(\tau;\beta)}\end{eqnarray} \end_inset Because \begin_inset Formula $\hat{\bfm{C}}(\tau;\beta)$ \end_inset is symmetric, \begin_inset Formula $\hat{\bfm{T}}(\tau;\beta)$ \end_inset will be reversible. \end_layout \begin_layout Subsection Bayesian estimate of transition probabilities from a single temperature \end_layout \begin_layout Standard \begin_inset ERT status inlined \begin_layout Standard %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% \end_layout \begin_layout Standard % APPLICATION \end_layout \begin_layout Standard %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% \end_layout \end_inset \end_layout \begin_layout Section Application to terminally-blocked alanine peptide \end_layout \begin_layout Standard \begin_inset LatexCommand \label{section:application-to-alanine-dipeptide} \end_inset \end_layout \begin_layout Itemize Description of system used in the test: solvated terminally-blocked alanine peptide, relevant details from the protocol \end_layout \begin_layout Itemize Comparison of transition probabilities as a function of temperature \end_layout \begin_layout Itemize Surprisingly, there is still a lot of information about transitions to/from states that haven't been sampled (or have been poorly sampled) from Bayesian approach at single temperature, but MLE or non-reversible estimate doesn't work very well. \end_layout \begin_layout Itemize Statistical error for reweighting generally smaller than that from single-temper ature Bayesian analysis \end_layout \begin_layout Itemize At low temperatures for poorly sampled transitions (e.g. 3->6), suspect reweighting may significantly underestimate uncertainty \end_layout \begin_layout Itemize Distributions of some transition probabilities at 302 K \end_layout \begin_layout Itemize Comparison of temperature dependence of timescales \end_layout \begin_layout Itemize Some figure showing how much different temperatures contribute to the estimates of particular transitions ? Extrapolation figure? \end_layout \begin_layout Standard \begin_inset ERT status inlined \begin_layout Standard %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% \end_layout \begin_layout Standard % DISCUSSION \end_layout \begin_layout Standard %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% \end_layout \end_inset \end_layout \begin_layout Section Discussion \end_layout \begin_layout Standard \begin_inset LatexCommand \label{section:discussion} \end_inset \end_layout \begin_layout Itemize Even at temperatures at which no transitions are observed, can still estimate transition probabilities. \end_layout \begin_layout Itemize Allows us to interpolate at temperatures that haven't been sampled; smoothly describe temperature-dependent behavior (fluctuations much smaller than Bayesian method) \end_layout \begin_layout Itemize We could even differentiate the estimates of transition probabilities if we wanted because trajectory weights are differentiable \end_layout \begin_layout Itemize Same is true for non-dynamic properties (expectations, stationary distributions) \end_layout \begin_layout Itemize Some transitions appear to be thermally activated processes (increasing probability with energy or temperature), others are rather flat \end_layout \begin_layout Itemize What about entropically-dominated barriers? In this case, heating would reduce rate, but because we can use lower temperatures, this can help too. \end_layout \begin_layout Itemize While temperature range at which other temperatures contribute significantly to weighting diminishes as system size increases, the requirement that temperatures in parallel tempering be spaced to get good exchange means that several replicas are always guaranteed to contribute \end_layout \begin_layout Itemize Any observations about metastable state structure or grouping as temperature changes? (PCCA+ as a function of temperature?) \end_layout \begin_layout Itemize What about the future? Some combination of Bayesian and reweighting (similar to Ron Levy's T-WHAM) may provide the best of both types of estimators in giving good uncertainties at the expense of introducing some bias from introduction of histograms in energy \end_layout \begin_layout Itemize Transition path sampling could help collect more data for poorly sampled transitions -- comment on "sweet spot" for energies of these transitions? \end_layout \begin_layout Itemize Limitations of requiring we use Hamiltonian trajectories -- could it work for longer trajectories or non-Hamiltonian dynamics? \end_layout \begin_layout Standard \begin_inset ERT status inlined \begin_layout Standard %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% \end_layout \begin_layout Standard % ACKNOWLEDGMENTS \end_layout \begin_layout Standard %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% \end_layout \end_inset \end_layout \begin_layout Section Acknowledgments \end_layout \begin_layout Standard \begin_inset LatexCommand \label{section:acknowledgments} \end_inset \end_layout \begin_layout Standard JDC and VSP gratefully acknowledge support from an NSF grant for Cyberinfrastruc ture (NSF CHE-0535616). \end_layout \begin_layout Standard \begin_inset ERT status inlined \begin_layout Standard %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% \end_layout \begin_layout Standard % BIBLIOGRAPHY \end_layout \begin_layout Standard %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset LatexCommand \bibtex[prsty]{markov-model-from-parallel-tempering} \end_inset \end_layout \begin_layout Standard \begin_inset ERT status inlined \begin_layout Standard %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% \end_layout \begin_layout Standard % APPENDICES \end_layout \begin_layout Standard %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% \end_layout \end_inset \end_layout \begin_layout Standard \start_of_appendix \begin_inset ERT status inlined \begin_layout Standard %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% \end_layout \begin_layout Standard % APPENDIX: PROOF THAT MODIFIED PARALLEL TEMPERING PROTOCOL GENERATES CORRECT DISTRIBUTION \end_layout \begin_layout Standard %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% \end_layout \end_inset \end_layout \begin_layout Section Proof that modified parallel tempering protocol generates canonical distribution \end_layout \begin_layout Standard \begin_inset LatexCommand \label{appendix:parallel-tempering-proof} \end_inset \end_layout \begin_layout Standard \begin_inset ERT status inlined \begin_layout Standard %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% \end_layout \begin_layout Standard % APPENDIX: PROPAGATION OF ERROR \end_layout \begin_layout Standard %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%% \end_layout \end_inset \end_layout \begin_layout Section Estimation of uncertainties in transition probabilities \end_layout \begin_layout Standard \begin_inset LatexCommand \label{appendix:propagation-of-error} \end_inset \end_layout \end_body \end_document