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Abstract Continuous-time Markov chains are frequently used as stochastic models for chemical reaction networks, especially in the growing field of systems biology. A fundamental problem for these Stochastic Chemical Reaction Networks (SCRNs) is to understand the dependence of the stochastic behavior of these systems on the chemical reaction rate parameters. Towards solving this problem, in this paper we develop theoretical tools called comparison theorems that provide stochastic ordering results for SCRNs. These theorems give sufficient conditions for monotonic dependence on parameters in these network models, which allow us to obtain, under suitable conditions, information about transient and steady-state behavior. These theorems exploit structural properties of SCRNs, beyond those of general continuous-time Markov chains. Furthermore, we derive two theorems to compare stationary distributions and mean first passage times for SCRNs with different parameter values, or with the same parameters and different initial conditions. These tools are developed for SCRNs taking values in a generic (finite or countably infinite) state space and can also be applied for non-mass-action kinetics models. When propensity functions are bounded, our method of proof gives an explicit method for coupling two comparable SCRNs, which can be used to simultaneously simulate their sample paths in a comparable manner. We illustrate our results with applications to models of enzymatic kinetics and epigenetic regulation by chromatin modifications. 

ABSTRACT We report the results of long-term reverberation mapping campaigns of the nearby active galactic nuclei (AGNs) NGC 4151, spanning from 1994 to 2022, based on archived observations of the FAST Spectrograph Publicly Archived Programs and our new observations with the 2.3 m telescope at the Wyoming Infrared Observatory. We reduce and calibrate all the spectra in a consistent way, and derive light curves of the broad H β line and 5100 Å continuum. Continuum light curves are also constructed using public archival photometric data to increase sampling cadences. We subtract the host galaxy contamination using Hubble Space Telescope imaging to correct fluxes of the calibrated light curves. Utilizing the long-term archival photometric data, we complete the absolute flux-calibration of the AGN continuum. We find that the H β time delays are correlated with the 5100 Å luminosities as $$\tau _{\rm H\beta }\propto L_{5100}^{0.46\pm 0.16}$$. This is remarkably consistent with Bentz et al. (2013)’s global size–luminosity relationship of AGNs. Moreover, the data sets for five of the seasons allow us to obtain the velocity-resolved delays of the H β line, showing diverse structures (outflows, inflows, and discs). Combining our results with previous independent measurements, we find the measured dynamics of the H β broad-line region (BLR) are possibly related to the long-term trend of the luminosity. There is also a possible additional ∼1.86 yr time lag between the variation in BLR radius and luminosity. These results suggest that dynamical changes in the BLR may be driven by the effects of radiation pressure.