\documentclass[10pt]{letter}
\address{
John D. Chodera\\
QB3 Institute\\
260J Stanley Hall \#3220\\
University of California\\
Berkeley, CA 94720-3220, USA \\
phone: 415.867.7384\\
email: \href{mailto:jchodera@berkeley.edu}{jchodera@berkeley.edu}\\
\name{John D. Chodera}
}

\signature{
\resizebox{2.5in}{!}{\includegraphics{JDC-signature}} \\
John D.~Chodera, corresponding author
} 

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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 are submitting a revised form of the following manuscript for consideration for publication as a {\bf full length article} in the Journal of the Chemical Physics:

{\bf Dynamical reweighting: Improved estimates of dynamical properties from simulations at multiple temperatures} by John D.~Chodera, William C.~Swope, Frank No\'{e}, Jan-Hendrik Prinz, Michael R.~Shirts, and Vijay S.~Pande.

We thank the referees for their positive and helpful comments on our earlier version of this manuscript, and have made revisions in accordance with their suggestions. 
Revisions due to referee request are marked in {\color{red}red} in the revised manuscript.
JCP also asked us to incorporate footnotes into the main body text; these are marked in {\color{blue}blue} in the revised manuscript.
We hope you find that the revised manuscript is now suitable for publication in JCP.

We ask that this manuscript be considered for back-to-back publication in the same issue with another manuscript concurrently in review at JCP [{\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}], since the included manuscript provides the theoretical groundwork for the estimator used in this companion paper.

We describe our changes in response to referee comments below.

\color{blue}
Reviewer \#1 Evaluations: \\
RECOMMENDATION: Publish in JCP with optional revision 

Reviewer \#1 (Comments to the Author): 

The authors describe a procedure to use re-weighting to combine information from various temperatures (or other control parameters) in order to optimize estimates of dynamical properties at the temperature of interest. This is an interesting extension of re-weighting techniques, usually used for calculating statical quantities, to the less obvious case of dynamic quantities. The method is described very clearly, and the presented results demonstrate conclusively its working.

\color{black}
We thank the referee for these kind comments.

\color{blue}
I found the manuscript at times too detailed in the method section. I suggest to streamline this section and move more technical parts such as the discussion of the thermostats into an appendix. 

\color{black}
We have considered the referee's suggestion carefully, but feel that the derivation of these thermostat path actions is an important primary contribution of the paper, and so have decided to leave the thermostat section in the main text.

\color{blue}
While the presentation of the presented data is very careful, and convinced me of the usefulness of the presented approach, I was a bit disappointed that interesting results such as transition rates along the $\Phi$ torsion is not shown. I would prefer if the authors could add at least some of these data instead of just pointing to a companion paper. I believe that this would help to increase the impact of this very interesting papers. 

\color{black}
We feel that the addition of a section demonstrating robust rate constant estimation for this example would add significant unnecessary additional technical complexity that would distract from the main point of the paper.
The main point of the paper is the precise estimation of temporally local trajectory functionals; the current example, in which a normalized fluctuation autocorrelation function is computed, conveys the concepts and advantages of our approach without bringing in the additional machinery of, for example, estimation of the rate constant by the reactive flux theory of Chandler [JCP 68:2959, 1978]. 
We also hope the companion paper, which uses the machinery of Markov models to estimate rates from finite-time transition probabilities and rates, will appear back-to-back with this theory paper in the journal, allowing interested readers to learn in detail how the theory here may be applied to more complex estimation problems.

\color{blue}
As a side remark: the redrawing the initial velocities from a Maxwell-Boltzmann distribution at the new temperature after an exchange move was introduced also in Ref. 6. 

\color{black}
We thank the referee for calling this to our attention.
A citation has been added.


\color{blue}
Reviewer \#2 Evaluations: \\
RECOMMENDATION: Publish in JCP with mandatory revision (minor) 

Reviewer \#2 (Comments to the Author): 

The paper by Chodera et al.~is a very interesting work concerned with advancing current methods for performing and analyzing molecular dynamics simulations at multiple temperatures. Notably, the authors demonstrate that the method is useful for optimal estimates of correlation functions in parallel tempering simulations, though the method seems to be general enough for being applied to other types of simulations. The paper is well written, the data presented supports the results and I expect it to be of interest to a significant fraction of the journal's readership. A few minor questions and suggestions are listed below. 

\color{black}
We thank the reviewer for his positive comments and taking the time to provide several helpful suggestions on how to improve our manuscript.

\color{blue}
1.- The notation related to Eqs. 10 and 11 is confusing. Even though these relations have been derived elsewhere, they are central to this paper and all their constants and variables should be defined here. 

\color{black}
We have added some explanatory text to make the variables used in these equations, and their interpretation, clearer.

\color{blue}
2.- As written, it is not clear how exactly do the authors evaluate the normalizing factors Z-hat in Eq. 11 (and of course 14) from actual trajectories. 

\color{black}
We now include a summary of one of the procedures for determining the normalizing factors (a simple iterative update procedure) in addition to citing our earlier work, which describes several procedures for the solution of these normalizing constants in detail.

\color{blue}
3.- What is the exact definition of the "bias" shown in fig. 3. Also, the two panels for the bias and variance are plotted too small. 

\color{black}
We have added a statement defining the statistical concepts of bias and variance as well as clarifying how they were estimated in this case.

\color{blue}
4.- For the alanine simulation, why did the authors use this specific number of replicas and this temperature range? How exactly where the replicas spaced exponentially in T? 

\color{black}
A clarifying statement has been added to explain these points.

\color{blue}
5.- One possible concern comes from the definition of the $h(x)$ function in Eq.~48. Such a simple definition for the $\alpha_R$ region could lead to Kramers-like recrossings that would make the evaluation of the correlation function in Eq 49 unreliable. The authors may want to discuss this aspect. 

\color{black}
While rate estimation can be tricky using a simple definition of $\alpha_R$---such as the one we have chosen---the correlation function for the defined function $h(x)$ is accurately estimated here, and appropriately contaminated by fast relaxation timescales.
Numerous rate theories---such as the reactive flux theory of Chandler [JCP 68:2959, 1978]---can provide rate estimates from these correlation functions that are robust to variations in the definition of a stable state.
We have revised our discussion of the choice of this definition to communicate this.

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