\documentclass[10pt]{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{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 revised our previously submitted draft 

{\bf Efficient Computation, Sensitivity and Error Analysis of Committor
Probabilities \\for Complex Dynamical Processes} \\ by Jan-Hendrik Prinz, Martin Held, Jeremy C. Smith and Frank No\'{e}.

regarding all comments of the reviewer. All changed paragraphs in have been marked \emph{red} in the draft for better readability. In the following we give a detailed explanation of the changes made in this revised version:

\begin{itemize}
  \item {page 2}
  \begin{quote}\small A technical point. I think it is possible to define $q(x)$ to be non-zero for $x \in \A$ if we think about transitions from $x \in \A$ that could find their way to $B$ without returning to $\A$ first. I think the definition is just a matter of convenience if the transitions are defined consistently. 
  \end{quote}
Yes, the committor theory can also be defined consistently with committor probabilities $q(x) > 0$ for states $x \in \A$. The ambiguity occurs only in time- or space-discretized dynamics and lies in the definition of when the hitting of set $\A$ is measured. To be precise, we state, that if we measure the committor of a state $x \in \A$ then it already hit set $\A$. Alternatively, one can choose to allow a first jump with a possibly non-zero probability to leave $\A$ and measure the committor probability $q(x)$ then, leading to a non-zero committor probability for states $x \in \A$. This choice does not affects states that are not in $\A$ or $\B$. To be consistent with the picture of a reaction coordinate from $\A \rightarrow \B$, we adopt the case of setting $q(x) \equiv 0, \forall x \in \A$ and $q(x) \equiv 1, \forall x\in B$. Similar arguments hold for states $x \in \B$. We have explained this on page 2 of the revised manuscript.

  \item {page 3}\begin{quote}\small It would be useful to make a connection with the rate constant expressions derived in terms of committors in Mol. Phys., 104, 1497, 2006 and Int. Rev. Phys. Chem., 25, 237, 2006. The expression in equation (9) is probably equivalent to one of these results.\end{quote}
Equation (9) had an error and has been corrected. The flux $F$ is the inverse of the total time for a cycle $\A\rightarrow \B \rightarrow \A$, which can be separated into the average times to travel from $\A$ to $\B$ expressed by the rate $k_{\A\B}$ and the other direction $k_{BA}$. The presented rate $k_{\A\B}$ is thus the inverse of the mean first passage time from $\A$ to $\B$. This is now derived on pages 4/5. 

Since our rate definition $k_{\A\B}$ of the \emph{milestoning} case can be expressed via mean first passage times which allow revisits, it is identical to the rate definitions of the KMC rates described in the papers [Mol. Phys. (2006) vol. 104 (9) pp. 1497-1507] or [Int. Rev. Phys. Chem. (2006) vol. 25 (1\&2) pp. 237 - 282], referred to as $k_{\A\B}^{\text{KMC}}$ and $k_{\B\A}^{\text{KMC}}$ in these publications. Since the calculation of rates is not the primary focus of our paper, we plan to investigate the relations between our approach and other rate definitions in a more detailed future work.
%To be precise, the rates can shown to be identical up to a normalization factor, due to slightly different definitions. While the TPT rate is the average number of trajectories traveling from $\A$ to $\B$ per time unit, the steady-state approximation, however, is a rate, used in a two-state master-equation model between states $\A$ and $\B$. In the latter case, the actual rate $k_{\A\B}^{TPT}$ of trajectories leaving $\A$ is expressed in this model by the probability of starting in state $\A$ times the \emph{local} (or conditional) rate $k_{AB}^{SS}$. This leads to the simply relation
%\begin{equation}
%	k^{SS}_{\A\B} = \frac{1}{\pi_{\A}} k^{TPT}_{\A\B}
%\end{equation}
%between both rates. This equality can then be shown by straight-forward algebraic transformations and acknowledging a slightly different definition of the committor as discussed in the paragraph above.
%I have extended the paragraph to include both rate definitions with additional citations.
	\item{page 5} \begin{quote}\small If $h^A$ is equivalent to a mean first passage time then this is probably worth mentioning. \end{quote}
$h^{A}$ is the hitting-time, which is the minimum number of steps to reach subset $A$ given a concrete realization of a discrete trajectory (or of a stochastic process) $X(\omega)$ out of all possible trajectories, that started at $X_{0} = x$. If we compute the expectation of the hitting time over all trajectories we get the mean first passage time
\[		\text{mfpt}(x)  =  \mathbb{E}_{X} \left( h^{A} (X) \mid X_{0} = x \right)  \]
	So $h^{A}$ and $\text{mfpt}(x)$ are different.
	
	\item{page 10ff} \begin{quote}\small My experience of iterative methods for calculating the committor probabilities, or eigenvalues of the transition matrix using Lanczos or Arnoldi-type methods, is that insurmountable numerical problems arise when the process of interest becomes slow. Here, ``slow'' means that the eigenvalue of interest becomes comparable to the precision of the zero eigenvalue of the transition matrix corresponding to the static equilibrium solution. The authors' approach certainly seems more efficient than these other iterative methods, but I wonder sensitive it is to this effect? Is there a clear separation in the eigenspectrum for the examples the authors consider? One particular advantage of the NGT procedure described in reference [34] is that precision is retained, even for very rare events. With the assumed dynamics in the two model systems it would probably be straightforward to compare the performance of the authors' approach with NGT, and this might be a useful avenue for future work. In the mean time, can the authors provide an estimate of how low kT can be compared to the highest overall barrier for their approach? \end{quote}
	We agree with the reviewers opinion, that numerical issues are indeed a big problem in the case of ``slow'' events. For the used Power Method, the speed of convergence depends on the spectral gap between the first two largest eigenvalues and is thus not optimized for the case of rare events, so that the reformulation into an eigenvector problem does not directly help with this issue. However, the simplicity allows for a memory efficient implementation with which very large systems can be treated, provided, that the spectral gap is large enough. This is presented in the 3D model, where probability can travel fast enough through the network since the diameter due to the cubic geometry is small ($<300$) and the transition probabilities are not too small. Thus convergence is slow, but assured. In the 2D example, the smaller size allows for a complete error and sensitivity analysis with medium high barriers ($\lambda_{2} = 0.999369$).
From a numerical point of view, a faster and more accurate convergence for large systems can be achieved using sparse matrix methods, such as Krylov space algorithms or graph based approaches like NGT[36]. Note, that besides the numerical view point, the reformulation of the committor problem in an eigenvector problem will allow this quantity to be related to the intrinsic dynamics of the system.
We extended the discussion about the power method by a short paragraph explaining its main area of applicability and the need of more advanced methods for dynamics with very slow processes.

	\item {page 26}\begin{quote}\small ``The method is efficently'' $\rightarrow$ ``The method is efficient'' \end{quote}
	Corrected!
\end{itemize}

Thank you for the helpful and interesting points of discussion!

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