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Abstract Continuous time Markov chains are commonly used as models for the stochastic behavior of chemical reaction networks. More precisely, these Stochastic Chemical Reaction Networks (SCRNs) are frequently used to gain a mechanistic understanding of how chemical reaction rate parameters impact the stochastic behavior of these systems. One property of interest is mean first passage times (MFPTs) between states. However, deriving explicit formulas for MFPTs can be highly complex. In order to address this problem, we first introduce the concept of$$coclique\, level\, structure$$and develop theorems to determine whether certain SCRNs have this feature by studying associated graphs. Additionally, we develop an algorithm to identify, under specific assumptions, all possible coclique level structures associated with a given SCRN. Finally, we demonstrate how the presence of such a structure in a SCRN allows us to derive closed form formulas for both upper and lower bounds for the MFPTs. Our methods can be applied to SCRNs taking values in a generic finite state space and can also be applied to models with non-mass-action kinetics. We illustrate our results with examples from the biological areas of epigenetics, neurobiology and ecology. 

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. 

Epigenetic cell memory, the inheritance of gene expression patterns across subsequent cell divisions, is a critical property of multicellular organisms. In recent work [S. Bruno, R. J. Williams, and D. Del Vecchio, PLOS Comput. Biol., 18 (2022), pp. 1–27], a subset of the authors observed in a simulation study how the stochastic dynamics and time scale differences between establishment and erasure processes in chromatin modifications (such as histone modifications and DNA methylation) can have a critical effect on epigenetic cell memory. In this paper, we provide a mathematical framework to rigorously validate and extend beyond these computational findings. Viewing our stochastic model of a chromatin modification circuit as a singularly perturbed, finite state, continuous time Markov chain, we extend beyond existing theory in order to characterize the leading coefficients in the series expansions of stationary distributions and mean first passage times. In particular, we characterize the limiting stationary distribution in terms of a reduced Markov chain, provide an algorithm to determine the orders of the poles of mean first passage times, and determine how changing erasure rates affects system behavior. The theoretical tools developed in this paper not only allow us to set a rigorous mathematical basis for the computational findings of our prior work, highlighting the effect of chromatin modification dynamics on epigenetic cell memory, but they can also be applied to other singularly perturbed Markov chains beyond the applications in this paper, especially those associated with chemical reaction networks. 

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.