{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,8]],"date-time":"2026-09-08T13:06:38Z","timestamp":1788872798016,"version":"build-2803163510"},"reference-count":44,"publisher":"AIP Publishing","issue":"10","funder":[{"DOI":"10.13039\/100000936","name":"Gordon and Betty Moore Foundation","doi-asserted-by":"publisher","award":["GBMF10790"],"award-info":[{"award-number":["GBMF10790"]}],"id":[{"id":"10.13039\/100000936","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["DE-SC0026333"],"award-info":[{"award-number":["DE-SC0026333"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["pubs.aip.org"],"crossmark-restriction":true},"short-container-title":[],"published-print":{"date-parts":[[2026,9,14]]},"abstract":"<jats:p>Molecular motors and other complex nonequilibrium systems are controlled by large sets of design parameters, and optimizing those parameters requires computing sensitivities\u2014derivatives of dynamical observables with respect to the parameters. When the system\u2019s dynamics involves rare events, both the observable and its sensitivity are difficult to estimate from direct simulation. We present a practical computational pipeline that addresses both challenges by combining importance sampling with a Markov state model (MSM). The MSM separately captures the slow, rare-event dynamics and the fast, local dynamics, and the chain rule connects those two pieces to yield an efficient sensitivity estimator. An iterative reweighting procedure based on the RiteWeight algorithm substantially reduces approximation errors from the MSM coarse-graining. We validate the approach on diffusion in the M\u00fcller\u2013Brown potential, where the sensitivity of a transition rate to landscape parameters can be computed exactly. We then use sensitivities to optimize the directional bias of a particle-based model of a catalysis-driven molecular motor.<\/jats:p>","DOI":"10.1063\/5.0345394","type":"journal-article","created":{"date-parts":[[2026,9,8]],"date-time":"2026-09-08T12:07:51Z","timestamp":1788869271000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":["Sensitivity analysis in the face of rare events"],"prefix":"10.1063","volume":"165","author":[{"ORCID":"https:\/\/orcid.org\/0009-0008-7542-8649","authenticated-orcid":false,"given":"John","family":"Strahan","sequence":"first","affiliation":[{"name":"Department of Chemistry, Northwestern University 1 , 2145 Sheridan Road, Evanston, Illinois 60208,","place":["USA"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4617-7146","authenticated-orcid":false,"given":"Todd R.","family":"Gingrich","sequence":"additional","affiliation":[{"name":"Department of Chemistry, Northwestern University 1 , 2145 Sheridan Road, Evanston, Illinois 60208,","place":["USA"]},{"name":"Department of Engineering Sciences and Applied Mathematics, Northwestern University 2 , 2145 Sheridan Road, Evanston, Illinois 60208,","place":["USA"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"317","published-online":{"date-parts":[[2026,9,8]]},"reference":[{"key":"2026090808073752900_c1","doi-asserted-by":"publisher","first-page":"e2024083118","DOI":"10.1073\/pnas.2024083118","article-title":"Designing self-assembling kinetics with differentiable statistical physics models","volume":"118","year":"2021","journal-title":"Proc. Natl. Acad. Sci. U. S. A."},{"key":"2026090808073752900_c2","doi-asserted-by":"publisher","first-page":"244901","DOI":"10.1063\/5.0130407","article-title":"Optimization of non-equilibrium self-assembly protocols using Markov state models","volume":"157","year":"2022","journal-title":"J.\u00a0Chem. Phys."},{"key":"2026090808073752900_c3","doi-asserted-by":"publisher","first-page":"2204","DOI":"10.1038\/s41467-022-29393-3","article-title":"Simulating a chemically fueled molecular motor with nonequilibrium molecular dynamics","volume":"13","year":"2022","journal-title":"Nat. Commun."},{"key":"2026090808073752900_c4","doi-asserted-by":"publisher","first-page":"e0256990","DOI":"10.1371\/journal.pone.0256990","article-title":"Differentiable molecular simulation can learn all the parameters in a coarse-grained force field for proteins","volume":"16","year":"2021","journal-title":"PLoS One"},{"key":"2026090808073752900_c5","unstructured":"W.\n            Wang\n          , S.Axelrod, and R.G\u00f3mez-Bombarelli, \u201cDifferentiable molecular simulations for control and learning,\u201d arXiv:2003.00868 (2020)."},{"key":"2026090808073752900_c6","first-page":"11428","article-title":"JAX MD: A framework for differentiable physics","volume":"33","year":"2020","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"2026090808073752900_c7","doi-asserted-by":"publisher","first-page":"e2426058122","DOI":"10.1073\/pnas.2426058122","article-title":"Reversible molecular simulation for training classical and machine-learning force fields","volume":"122","year":"2025","journal-title":"Proc. Natl. Acad. Sci. U. S. A."},{"key":"2026090808073752900_c8","doi-asserted-by":"publisher","first-page":"041032","DOI":"10.1103\/physrevx.13.041032","article-title":"Optimal control of nonequilibrium systems through automatic differentiation","volume":"13","year":"2023","journal-title":"Phys. Rev. X"},{"key":"2026090808073752900_c9","doi-asserted-by":"publisher","first-page":"e2508255123","DOI":"10.1073\/pnas.2508255123","article-title":"Fitting coarse-grained models to macroscopic experimental data via automatic differentiation","volume":"123","year":"2026","journal-title":"Proc. Natl. Acad. Sci. U. S. A."},{"key":"2026090808073752900_c10","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1145\/84537.84552","article-title":"Likelihood ratio gradient estimation for stochastic systems","volume":"33","year":"1990","journal-title":"Commun. ACM"},{"key":"2026090808073752900_c11","doi-asserted-by":"publisher","first-page":"034103","DOI":"10.1063\/1.3280166","article-title":"Efficient computation of parameter sensitivities of discrete stochastic chemical reaction networks","volume":"132","year":"2010","journal-title":"J.\u00a0Chem. Phys."},{"key":"2026090808073752900_c12","doi-asserted-by":"publisher","first-page":"034115","DOI":"10.1063\/1.3677230","article-title":"A pathwise derivative approach to the computation of parameter sensitivities in discrete stochastic chemical systems","volume":"136","year":"2012","journal-title":"J.\u00a0Chem. Phys."},{"key":"2026090808073752900_c13","doi-asserted-by":"publisher","first-page":"311","DOI":"10.1186\/1471-2105-14-311","article-title":"Parametric sensitivity analysis for biochemical reaction networks based on pathwise information theory","volume":"14","year":"2013","journal-title":"BMC Bioinf."},{"key":"2026090808073752900_c14","doi-asserted-by":"publisher","first-page":"2237","DOI":"10.1137\/110849079","article-title":"An efficient finite difference method for parameter sensitivities of continuous time Markov chains","volume":"50","year":"2012","journal-title":"SIAM J. Numer. Anal."},{"key":"2026090808073752900_c15","doi-asserted-by":"publisher","first-page":"2386","DOI":"10.1021\/jacs.7b12191","article-title":"Markov state models: From an art to a science","volume":"140","year":"2018","journal-title":"J.\u00a0Am. Chem. Soc."},{"key":"2026090808073752900_c16","doi-asserted-by":"publisher","first-page":"174105","DOI":"10.1063\/1.3565032","article-title":"Markov models of molecular kinetics: Generation and validation","volume":"134","year":"2011","journal-title":"J.\u00a0Chem. Phys."},{"key":"2026090808073752900_c17","doi-asserted-by":"publisher","first-page":"146","DOI":"10.1006\/jcph.1999.6231","article-title":"A direct approach to conformational dynamics based on hybrid Monte Carlo","volume":"151","year":"1999","journal-title":"J.\u00a0Comput. Phys."},{"key":"2026090808073752900_c18","doi-asserted-by":"publisher","first-page":"244111","DOI":"10.1063\/1.5063730","article-title":"Galerkin approximation of dynamical quantities using trajectory data","volume":"150","year":"2019","journal-title":"J.\u00a0Chem. Phys."},{"key":"2026090808073752900_c19","doi-asserted-by":"publisher","first-page":"094115","DOI":"10.1063\/5.0098587","article-title":"Augmented transition path theory for sequences of events","volume":"157","year":"2022","journal-title":"J.\u00a0Chem. Phys."},{"key":"2026090808073752900_c20","doi-asserted-by":"publisher","first-page":"715","DOI":"10.1137\/s0895479892228900","article-title":"Sensitivity of the stationary distribution of a Markov chain","volume":"15","year":"1994","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"2026090808073752900_c21","doi-asserted-by":"publisher","first-page":"917","DOI":"10.1137\/140987900","article-title":"Sharp entrywise perturbation bounds for Markov chains","volume":"36","year":"2015","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"2026090808073752900_c22","doi-asserted-by":"publisher","first-page":"e2529246123","DOI":"10.1073\/pnas.2529246123","article-title":"Randomized iterative trajectory reweighting for steady-state distributions without discretization error","volume":"123","year":"2026","journal-title":"Proc. Natl. Acad. Sci. U. S. A."},{"key":"2026090808073752900_c23","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1007\/BF00547608","article-title":"Location of saddle points and minimum energy paths by a constrained simplex optimization procedure","volume":"53","year":"1979","journal-title":"Theor. Chim. Acta"},{"key":"2026090808073752900_c24","first-page":"1","article-title":"Transition path sampling","volume-title":"Advances in Chemical Physics","year":"2002"},{"key":"2026090808073752900_c25","doi-asserted-by":"publisher","first-page":"229","DOI":"10.1023\/a:1022672621406","article-title":"Simple statistical gradient-following algorithms for connectionist reinforcement learning","volume":"8","year":"1992","journal-title":"Mach. Learn."},{"key":"2026090808073752900_c26","volume-title":"Stochastic Differential Equations: An Introduction with Applications","year":"2003","edition":"6"},{"key":"2026090808073752900_c27","doi-asserted-by":"publisher","first-page":"011066","DOI":"10.1103\/physrevx.10.011066","article-title":"Universal thermodynamic bounds on nonequilibrium response with biochemical applications","volume":"10","year":"2020","journal-title":"Phys. Rev. X"},{"key":"2026090808073752900_c28","doi-asserted-by":"publisher","first-page":"044113","DOI":"10.1103\/physreve.108.044113","article-title":"Topologically constrained fluctuations and thermodynamics regulate nonequilibrium response","volume":"108","year":"2023","journal-title":"Phys. Rev. E"},{"key":"2026090808073752900_c29","doi-asserted-by":"publisher","first-page":"037101","DOI":"10.1103\/physrevlett.132.037101","article-title":"Nonequilibrium response for Markov jump processes: Exact results and tight bounds","volume":"132","year":"2024","journal-title":"Phys. Rev. Lett."},{"key":"2026090808073752900_c30","doi-asserted-by":"publisher","first-page":"107103","DOI":"10.1103\/physrevlett.133.107103","article-title":"General theory of static response for Markov jump processes","volume":"133","year":"2024","journal-title":"Phys. Rev. Lett."},{"key":"2026090808073752900_c31","doi-asserted-by":"publisher","first-page":"L012103","DOI":"10.1103\/scg2-qkxv","article-title":"Universal response inequalities beyond steady states via trajectory information geometry","volume":"112","year":"2025","journal-title":"Phys. Rev. E"},{"key":"2026090808073752900_c32","doi-asserted-by":"publisher","first-page":"1507","DOI":"10.1214\/19-aap1468","article-title":"Sensitivity analysis for rare events based on R\u00e9nyi divergence","volume":"30","year":"2020","journal-title":"Ann. Appl. Probab."},{"key":"2026090808073752900_c33","doi-asserted-by":"publisher","first-page":"104107","DOI":"10.1063\/1.4943388","article-title":"Efficient estimators for likelihood ratio sensitivity indices of complex stochastic dynamics","volume":"144","year":"2016","journal-title":"J.\u00a0Chem. Phys."},{"key":"2026090808073752900_c34","doi-asserted-by":"publisher","first-page":"2959","DOI":"10.1063\/1.436049","article-title":"Statistical mechanics of isomerization dynamics in liquids and the transition state approximation","volume":"68","year":"1978","journal-title":"J.\u00a0Chem. Phys."},{"key":"2026090808073752900_c35","doi-asserted-by":"publisher","first-page":"391","DOI":"10.1146\/annurev.physchem.040808.090412","article-title":"Transition-path theory and path-finding algorithms for the study of rare events","volume":"61","year":"2010","journal-title":"Annu. Rev. Phys. Chem."},{"key":"2026090808073752900_c36","first-page":"81257","article-title":"The surprising efficiency of temporal difference learning for rare event prediction","volume":"37","year":"2024","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"2026090808073752900_c37","doi-asserted-by":"publisher","first-page":"134105","DOI":"10.1063\/5.0057323","article-title":"Reinforcement learning of rare diffusive dynamics","volume":"155","year":"2021","journal-title":"J.\u00a0Chem. Phys."},{"key":"2026090808073752900_c38","doi-asserted-by":"publisher","first-page":"e2210500120","DOI":"10.1073\/pnas.2210500120","article-title":"Sterically driven current reversal in a molecular motor model","volume":"120","year":"2023","journal-title":"Proc. Natl. Acad. Sci. U. S. A."},{"key":"2026090808073752900_c39","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1021\/acs.jctc.3c01201","article-title":"Limits on the precision of catenane molecular motors: Insights from thermodynamics and molecular dynamics simulations","volume":"20","year":"2024","journal-title":"J.\u00a0Chem. Theory Comput."},{"key":"2026090808073752900_c40","doi-asserted-by":"publisher","first-page":"1063","DOI":"10.1021\/jacs.4c14481","article-title":"Power strokes in molecular motors: Predictive, irrelevant, or somewhere in between?","volume":"147","year":"2025","journal-title":"J.\u00a0Am. Chem. Soc."},{"key":"2026090808073752900_c41","article-title":"It takes two to make a thing go right: Boosting current in coupled motors","year":"2026","journal-title":"J. Chem. Phys."},{"key":"2026090808073752900_c42","doi-asserted-by":"publisher","first-page":"024116","DOI":"10.1063\/1.2953328","article-title":"Measurement of nonequilibrium entropy from space-time thermodynamic integration","volume":"129","year":"2008","journal-title":"J.\u00a0Chem. Phys."},{"key":"2026090808073752900_c43","volume-title":"Molecular Dynamics: With Deterministic and Stochastic Numerical Methods","year":"2015"},{"key":"2026090808073752900_c44","doi-asserted-by":"publisher","first-page":"3837","DOI":"10.1038\/s41467-019-11402-7","article-title":"Kinetic asymmetry allows macromolecular catalysts to drive an information ratchet","volume":"10","year":"2019","journal-title":"Nat. Commun."}],"container-title":["The Journal of Chemical Physics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/pubs.aip.org\/aip\/jcp\/article-pdf\/doi\/10.1063\/5.0345394\/21130333\/104106_1_5.0345394.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/pubs.aip.org\/aip\/jcp\/article-pdf\/doi\/10.1063\/5.0345394\/21130333\/104106_1_5.0345394.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,9,8]],"date-time":"2026-09-08T12:07:53Z","timestamp":1788869273000},"score":1,"resource":{"primary":{"URL":"https:\/\/pubs.aip.org\/jcp\/article\/165\/10\/104106\/3403925\/Sensitivity-analysis-in-the-face-of-rare-events"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,9,8]]},"references-count":44,"journal-issue":{"issue":"10","published-print":{"date-parts":[[2026,9,14]]}},"URL":"https:\/\/doi.org\/10.1063\/5.0345394","relation":{},"ISSN":["0021-9606","1089-7690"],"issn-type":[{"value":"0021-9606","type":"print"},{"value":"1089-7690","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2026,9,14]]},"published":{"date-parts":[[2026,9,8]]},"article-number":"104106"}}