PHYSICAL REVIEW LETTERS 122, 059802 (2019) de la Cruz et al. Reply In our recent Letter [1], we study Applying the Kolmogorov-Smirnov test to the resulting the transitions out of an oscillatory state for stochastic empirical cumulative distribution functions, the test cannot systems that can be described with a chemical Langevin reject the null hypothesis that the distributions are equal at a equation (CLE). As Meerson and Smith point out in the significance level α ¼ 0.05. Additionally, in the Letter we preceding Comment [2], Langevin equations are sometimes predicted the minimal action path using three different derived using the van Kampen expansion to the leading methods (minimizing the action, simulating the master order in the inverse of the population size in the master equation, and solving the CLE). We think it is extremely equation (ME) [3]. We rather derive the CLE using results unlikely that these three completely different methods dating back to early work by Kurtz regarding the Central will yield matching numerical artefacts, as Smith and Limit Theorem for Markov chains [4–6] and later by Meerson imply. Gillespie [7]. The resulting CLE we use has multiplicative Smith and Meerson point to references on the topic noise and is not recovered, even using the generalized van [9–13]. While relevant, none of these references deal Kampen expansion [8]. The validity of this derivation does with a comparable dynamical landscape to the one in our not rely on large system-size asymptotics, and its higher Letter (in which there is a stable and an unstable limit accuracy was analyzed by Grima et al. [8] for the first two cycle). In these Letters, the dynamics are affected by a moments of the distribution. It requires the existence of a saddle point or a focus, while our Letter deals with the timescale on which each reaction channel is triggered transition from a limit cycle crossing an unstable limit multiple times and the propensity of each reaction stays cycle without saddles. Additionally, all of these refer- approximately constant [7]. To verify that such a condition ences make use of additive noise, while one of the is met in the cases discussed in our Letter, we have run essential aspects of the CLE is the multiplicative noise. exact stochastic simulation algorithm (SSA) simulations of Finally, we regret that we did not cite Smith and Meerson the ME. The results are shown in Fig. 1, where, on the [14]. Since they look at extinctions, the CLE approach timescale of τ = 10 − 5 , each reaction channel is triggered at will not be accurate in their case, since death or least 100 times, and the maximal relative change in a degradation propensities will become negligible close reaction rate is less than 10%. to the extinction point. They suggest the use of the WKB To show the accuracy of the CLE to describe the method in this case [15], which is very accurate in transition between a stable oscillation and a fixed point, predicting the quasistable distribution. The CLE, while we compare the first passage time distributions from direct more accurate than the van Kampen Fokker Plank Gillespie SSA simulations of the ME and numerical equation, was inaccurate in the tails of the distributions, solutions of our CLE. The results are shown in Fig. 1. where all of the probabilities are vanishingly small. FIG. 1. (Left) Escape trajectories for three different realizations of the ME. (Top left) Minimum number of times that one of the four reactions occurs (ni ) in windows of time τ ¼ 10−5 along the escape trajectory. (Inset) Trajectories for the three different realizations. (Bottom left) Maximum relative change of the propensity (π i ) among all the possible reactions in windows of time τ ¼ 10−5 . (Right) cumulative distribution functions of the First Passage Time for different sets of 600 jumps of the ME and the CLE. Confidence bands are created using the Dvoretzky-Kiefer-Wolfowitz inequality (confidence of 99%). Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. 0031-9007=19=122(5)=059802(2) 059802-1 Published by the American Physical Society PHYSICAL REVIEW LETTERS 122, 059802 (2019) R. dlC. acknowledges AGAUR for funding under its Received 11 July 2018; published 8 February 2019 doctoral scholarship program. R. dlC. and T. A. are sup- DOI: 10.1103/PhysRevLett.122.059802 ported by the CERCA Program of the Generalitat de Catalunya, MINECO (Grants No. MTM2015-71509-C2- 1-R and No. CRM-BGSMath “María de Maeztu” program * No. MDM-2014-0445), and AGAUR (Grant Corresponding author. No. 2014SGR1307). R. P-C., P. G., and K. M. P. were r.carrasco@ucl.ac.uk [1] R. de la Cruz, R. Perez-Carrasco, P. Guerrero, T. Alarcon, supported by the Wellcome Trust (Grant and K. M. Page, Phys. Rev. Lett. 120, 128102 (2018). No. WT098325MA). R. P-C. also acknowledges the [2] B. Meerson and N. R. Smith, arXiv:1804.01173v1. 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