\documentclass[11pt]{letter}
\address{
Jan-Hendrik Prinz\\
Computational Molecular Biology\\
Arnimallee 6\\
FU Berlin\\
14195 Berlin, Germany \\
phone: +49 (0)30 838 75775\\
email: \href{mailto:jan-hendrik.prinz@fu-berlin.de}{jan-hendrik.prinz@fu-berlin.de}\\
\name{Jan-Hendrik Prinz}
}

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Jan-Hendrik Prinz
} 

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\begin{document}
\begin{letter}{
Marsha I. Lester, Editor\\
\emph{The Journal of Chemical Physics}\\
Department of Chemistry\\
University of Pennsylvania\\
231 S.~34th St.\\
Philadelphia, PA 19104-6323, USA
}

\opening{Dear Editor,}

We have extensively revised our previously submitted draft 

{\bf Optimal use of data in parallel tempering simulations for the construction of discrete-state Markov models of biomolecular dynamics} \\ by Jan-Hendrik Prinz, John D. Chodera, Vijay S. Pande, William C. Swope, Jeremy C. Smith and Frank No\'{e}.

in response to the the reviewer comments and to improve the overall clarity and readability of the manuscript. 
Substantially modified paragraphs in the revised manuscript have been marked in {\color{red}\emph{red}} to facilitate re-review.  
We address the reviewer comments below, reprinting the original comments as appropriate.
\vspace{0.5cm}

{\bf \large First review}

We thank the reviewer for the thorough and careful discussion of the content and its presentation. 
We have addressed the stated concerns and comments in this revision, and in the following we give a detailed explanation of changes made in response to the reviewer's suggestion.

\begin{itemize}
\item {}
  \begin{quote}\small The notation is confusing and inconsistent. For example, two indices are used without justification for the trajectories $z_{nk}$ (page 9 line 5) and only one index later on for $z_n$ (page 9 line 11). What is the difference between the $n$ and $k$ indices in the first case? What is $T$ in this paragraph (presumably the total time of one(?) trajectory, but it is undefined in the text, the same symbol being used for temperature)? Similarly, too many indices seem to be used for beta in the paragraph between equations 18 and 19, and further on in the paper.   
  \end{quote}
  We agree with the reviewer that our original notation was unnecessary complex.
  We have, throughout all theory sections, revised the notation and removed unnecessary notation. 
  We have taken special care to ensure all symbols used are now clearly defined.
\item {}
    \begin{quote}\small The trajectory weights ``$w$'' used first in Eq. 20 are not defined and introduced properly. Equations 20, 21, and 22, which are central to this paper, are neither discussed nor justified here. There is a vague reference to Ref. 32 but that paper is listed as "in preparation", so all the information about the weights should be also clearly included here. 
    \end{quote}
    Our intention was to publish this paper side-by-side with the companion theory paper, which is cited in the paper as the reference for a more extensive presentation of the \emph{dynamical reweighting} method. 
    (In fact, we asked the editor to send these papers out for joint review.)
    The companion theory paper is now accepted and the citation in the bibliography has been updated. 
    Nonetheless, we have revised Section II of this manuscript to contain a brief but complete outline to allow the reader to understand and apply the method even without referring to the companion paper. 
    The companion theory paper contains detailed derivations, proofs, and extensions to a variety of ensembles, which we refer the reader to for more detail on these aspects.
\item{}
    \begin{quote} \small In Eq. 23, if $k$ is used for indexing the trajectories by temperature, what is the index $n$ used for? 
    \end{quote}
    We have corrected this in the revision.
\item {}
    \begin{quote} \small What exactly is $\tau$ used as in $T \ge \tau$ on page 11 line 14? 
    \end{quote}
    The symbol $\tau$ always refers to lagtime throughout the paper. 
    We have included additional explanations, and where necessary, also changed the symbol for trajectory length to avoid confusion.
\item {}
    \begin{quote} \small The justification of reassignment of velocities instead of rescaling (page 12 line 13) is unclear. \end{quote}
    We have clarified this in the revised text.
    In brief, because we use Hamiltonian trajectories between exchanges, velocity reassignment (or some procedure to the same effect) is necessary to ensure proper thermostatting of the individual temperatures in the parallel tempering simulation.
%    The reason for the reassignment is that it decreases the correlation times between trajectory parts of neighboring temperatures. 
%    This leads to a better error estimate and a faster conversion, which was the main reason to use parallel tempering in the first place. 
%    It is important to note, that this is useful for \emph{dynamical reweighting} but not necessary and in general often desirable for a PT simulation.
\item {}
    \begin{quote} \small It is not clear how much of the data generated here is new, and how much has been published already in different papers such as Ref. 31 and 32.
    \end{quote}
    We have clarified that we utilized data generated for an earlier study to facilitate comparison to earlier work, in which a Markov model was shown to accurately describe long-time dynamics for the alanine dipeptide system we use as an illustration of the methodology.
    \item {}
    \begin{quote} \small In Fig. 3, the authors compare DR data from all temperatures to TE data for each temperature. However, the TE data is based on much shorter (the number of replicas, $N_k$ times shorter) total simulation times, as they used only each trajectory from each temperature. Arguably, a fair comparison should compare data obtained for the same total "computational resources". That is, the authors should analyze as much data for each single temperature as for all the replica exchange trajectories. 
    \end{quote}
    The goal of this work is to show that the use of dynamical reweighting can produce more precise temperature-dependent estimates of Markov models from parallel tempering trajectories than simply using the replica trajectories that visit a single temperature (the scheme of Buchete and Hummer).  
    The referee's recommendation would absolutely make sense if we were interested in only a single temperature, but we focus on temperature-dependent kinetics in this paper, and so this comparison would be inappropriate.
\item {}
    \begin{quote} \small Fig. 4 and its discussion are also confusing. First, the y-scale for transition "probabilities" runs from 0 to 10! Also, as stated above, comparing single temperature data for much shorter total trajectories to the large DR and ST data sets seems unfair. 
    \end{quote}
Fig.~4 shows the posterior probability density function $p(T_{i\rightarrow j})$ of the transition matrix elements $T_{i \rightarrow j}$. 
Recall that probability density functions are nonnegative functions that integrate to unity; there is no requirement that $p(T)$ cannot exceed unity for some interval of $T$.
While we are interested in temperature-dependent properties, the figure compares multiple approaches to estimating the transition probabilities at a single temperature for clarity, and because a highly accurate estimate of the transition probabilities (the ``reference'') is available from another dataset generated with a different technique.
We have attempted to clarify these issues in the figure caption and in the text.
\item {}
    \begin{quote} \small In Fig. 6 the color scale may be misleading when comparing actual values, as it is basically a binary (red for values $> 60\%$ and yellow/white for values $< 60\%$ ). Why not use a linear gray scale? Also, what exactly does it mean a certain value of $x\%$ of the intensity in the color bar? 
    \end{quote}
    We thank the referee for this suggestion.
    We have changed to grayscale and revised the caption and text describing the comparison of transition matrices using a scalar product similarity measure.
\item {}
    \begin{quote} \small In Eq. 37, 38 and 39 (!), how could a matrix, $T^\text{sym}$, be equal to a number (as appears to be the sum over the elements of $Q_i$) in Eq. 39? It is definitely some wrong notation. 
    \end{quote}
The intended notation is correct and represents the sum of the rank-one matrices $Q_{i}$. 
To make things clearer, we have revised this paragraph and reduced the number of symbols employed.
	\item {}
    \begin{quote} \small The ``argmax'' and ``Sim'' notations in Eqs. 11 and 40 are confusing. \end{quote}
We have made edits to avoid confusion.
	\item {}
    \begin{quote} \small In Fig. 7a , it is shown that only data from neighboring temperatures is really used for estimates at a certain temperature. That seems to contradict the need to perform this kind of PT simulations since one of the goals of replica exchange is to use data from temperatures as far away from the temperature of interest as feasible in order to enhance the sampling at the target temperature of interest.
    \end{quote}
There is, in fact, no contradiction.
Parallel tempering is generally employed for two reasons: \emph{(1)} to reduce correlation times by allowing the system to heat up and cool down, thus often requiring less wall clock time to generate the same number of uncorrelated samples than multiple simulations at a single temperature; and \emph{(2)} to study temperature-dependent properties.
When we analyze the parallel tempering simulation with dynamical reweighting, as in the current manuscript, we \emph{still} have both advantages of parallel tempering.
The reweighting process draws most of the information about the temperature of interest from a few neighboring temperatures for reasons of phase space overlap, but we still have the advantage that there are more uncorrelated trajectories and we can extract temperature-dependent dynamics compared to having run simulations at a \emph{single} temperature.
	\item {}
    \begin{quote} \small What is the meaning of \emph{counts} in Fig. 7b, because the color scales shows values between 0 and 1!
    \end{quote}
    This represents the effective relative weight that trajectories sampled from one temperature receive at another temperature of interest, with trajectories sampled from the temperature of interest receiving a relative weight of unity.
    We realize that our previous presentation was confusing, and have clarified this in the text and figure caption.
    	\item {}
    \begin{quote} \small Some references are wrong, duplicated. For example, compare Refs. 28 and 46, 31 and 45, 38 and 47, etc. 
    \end{quote}
	It appears this error was caused by an incomplete merging of two bibliographies.
	We have corrected this in the updated version.

\end{itemize}

\vspace{0.5cm}
{\bf \large Second review}

We thank the reviewer for the kind comments, his opinion on the presented material, and interesting points of discussion.
We have carefully addressed the comments and concerns in the following:

\begin{itemize}
  \item {}
  \begin{quote}\small This sentence seems out of line:
  
  ``The degree to which the use of PT can enhance the thermodynamic sampling efficiency is limited'' 
  because quite a bit of work has shown PT to better equilibrate a variety of systems. 
  \end{quote}
  We agree with the reviewer that indeed PT is an excellent way to enhance the speed of equilibration in the simulated system, but simply wanted to point out that PT is not a ``magical cure-all'' that eliminates all sampling challenges---therefore, the degree to which it can enhance sampling is limited.
  We have clarified our discussion of this point in the revised text.
  \item {}
  \begin{quote}\small 
  Since the overestimation of the error was mentioned several times: perhaps the authors should construct the error from a non-Gaussian prior.
  \end{quote}
  We agree with the reviewer, that non-gaussian priors would lead to much better estimates for the presented method. The main reason for using a gaussian approximation in DR is, that this can be propagated in an analytically form and finally be expressed in a closed form. A non-gaussian prior can either be propagated in a few cases by a more complicated mathematical scheme or, as used by the other two methods (TE) and (RE), by sampling. This might be a point for further investigations.
\end{itemize}

We thank both reviewers for the interesting and very helpful points of discussion!

\closing{Kind regards,}
\end{letter}
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