In our recent Letter, we study the transitions out of an oscillatory state for stochastic systems that can be described with a chemical Langevin equation (CLE). This is a reply to comments on that letter

Alarcon, T

de la Cruz, R

Guerrero, P

Page, KM

Perez-Carrasco, R

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de la Cruz et al. Reply In our recent Letter [1], we studythe transitions out of an oscillatory state for stochasticsystems that can be described with a chemical Langevinequation (CLE). As Meerson and Smith point out in thepreceding Comment [2], Langevin equations are sometimesderived using the van Kampen expansion to the leadingorder in the inverse of the population size in the masterequation (ME) [3]. We rather derive the CLE using resultsdating back to early work by Kurtz regarding the CentralLimit Theorem for Markov chains [4–6] and later byGillespie [7]. The resulting CLE we use has multiplicativenoise and is not recovered, even using the generalized vanKampen expansion [8]. The validity of this derivation doesnot rely on large system-size asymptotics, and its higheraccuracy was analyzed by Grima et al. [8] for the first twomoments of the distribution. It requires the existence of atimescale on which each reaction channel is triggeredmultiple times and the propensity of each reaction staysapproximately constant [7]. To verify that such a conditionis met in the cases discussed in our Letter, we have runexact stochastic simulation algorithm (SSA) simulations ofthe ME. The results are shown in Fig. 1, where, on thetimescale of τ = 10− 5, each reaction channel is triggered atleast 100 times, and the maximal relative change in areaction rate is less than 10%.To show the accuracy of the CLE to describe thetransition between a stable oscillation and a fixed point,we compare the first passage time distributions from directGillespie SSA simulations of the ME and numericalsolutions of our CLE. The results are shown in Fig. 1.Applying the Kolmogorov-Smirnov test to the resultingempirical cumulative distribution functions, the test cannotreject the null hypothesis that the distributions are equal at asignificance level α ¼ 0.05. Additionally, in the Letter wepredicted the minimal action path using three differentmethods (minimizing the action, simulating the masterequation, and solving the CLE). We think it is extremelyunlikely that these three completely different methodswill yield matching numerical artefacts, as Smith andMeerson imply.Smith and Meerson point to references on the topic[9–13]. While relevant, none of these references dealwith a comparable dynamical landscape to the one in ourLetter (in which there is a stable and an unstable limitcycle). In these Letters, the dynamics are affected by asaddle point or a focus, while our Letter deals with thetransition from a limit cycle crossing an unstable limitcycle without saddles. Additionally, all of these refer-ences make use of additive noise, while one of theessential aspects of the CLE is the multiplicative noise.Finally, we regret that we did not cite Smith and Meerson[14]. Since they look at extinctions, the CLE approachwill not be accurate in their case, since death ordegradation propensities will become negligible closeto the extinction point. They suggest the use of the WKBmethod in this case [15], which is very accurate inpredicting the quasistable distribution. The CLE, whilemore accurate than the van Kampen Fokker Plankequation, was inaccurate in the tails of the distributions,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 fourreactions 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 (πi) 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. Confidencebands 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. Furtherdistribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.PHYSICAL REVIEW LETTERS 122, 059802 (2019)0031-9007=19=122(5)=059802(2) 059802-1 Published by the American Physical Society R. dlC. acknowledges AGAUR for funding under itsdoctoral scholarship program. R. dlC. and T. A. are sup-ported by the CERCA Program of the Generalitat deCatalunya, MINECO (Grants No. MTM2015-71509-C2-1-R and No. CRM-BGSMath “María de Maeztu” programNo. MDM-2014-0445), and AGAUR (GrantNo. 2014SGR1307). R. P-C., P. G., and K.M. P. weresupported by the Wellcome Trust (GrantNo. WT098325MA). R. P-C. also acknowledges theUCL Mathematics Clifford Fellowship.R. dlC. andR. P-C. contributed equally to this work.Roberto de la Cruz,1,2 Ruben Perez-Carrasco,3,*Pilar Guerrero,3 Tomas Alarcon1,2,4,5 and Karen M. Page31Centre de Recerca Matema`ticaEdifici C, Campus de Bellaterra08193 Bellaterra, Barcelona, Spain2Departament de Matema`tiquesUniversitat Autònoma de Barcelona08193 Bellaterra, Barcelona, Spain3Department of Mathematics, University College LondonGower Street, London WC1E 6BT, UK4ICREAPg. Lluís Companys 23, 08010 Barcelona, Spain5Barcelona Graduate School of Mathematics (BGSMath)Barcelona, SpainReceived 11 July 2018; published 8 February 2019DOI: 10.1103/PhysRevLett.122.059802*Corresponding author.r.carrasco@ucl.ac.uk[1] R. de la Cruz, R. Perez-Carrasco, P. Guerrero, T. Alarcon,and K. M. Page, Phys. Rev. Lett. 120, 128102 (2018).[2] B. Meerson and N. R. Smith, arXiv:1804.01173v1.[3] N. G. V. Kampen, Stochastic Processes in Physics andChemistry (Elsevier, New York, 2007).[4] T. G. Kurtz, Stoch. Proc. Appl. 6, 223 (1978).[5] T. G. Kurtz, Ann. Probab. 9, 557 (1981).[6] D. F. Anderson and T. G. Kurtz, Stochastic Analysis ofBiochemical Systems (Springer, New York, 2010).[7] D. T. Gillespie, J. Chem. Phys. 113, 297 (2000).[8] R. Grima and P. Thomas, and A. V. Straube, J. Chem. Phys.135, 084103 (2011).[9] R. Kautz, Phys. Lett. A 125, 315 (1987).[10] R. Kautz, Phys. Rev. A 38, 2066 (1988).[11] P. Grassberger, J. Phys. A 22, 3283 (1989).[12] R. Kautz, Rep. Prog. Phys. 59, 935 (1996).[13] V. N. Smelyanskiy, M. I. Dykman, and R. S. Maier, Phys.Rev. E 55, 2369 (1997).[14] N. R. Smith and B. Meerson, Phys. Rev. E 93, 032109(2016).[15] M. Assaf and B. Meerson, Phys. Rev. E 75, 031122 (2007).PHYSICAL REVIEW LETTERS 122, 059802 (2019)059802-2

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