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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 In the last decade, the interactions among histone modifications and DNA methylation and their effect on the DNA structure, i.e., chromatin state, have been identified as key mediators for the maintenance of cell identity, defined as epigenetic cell memory. In this paper, we determine how the positive feedback loops generated by the auto- and cross-catalysis among repressive modifications affect the temporal duration of the cell identity. To this end, we conduct a stochastic analysis of a recently published chromatin modification circuit considering two limiting behaviors: fast erasure rate of repressive histone modifications or fast erasure rate of DNA methylation. In order to perform this mathematical analysis, we first show that the deterministic model of the system is a singular singularly perturbed (SSP) system and use a model reduction approach for SSP systems to obtain a reduced one-dimensional model. We thus analytically evaluate the reduced system’s stationary probability distribution and the mean switching time between active and repressed chromatin states. We then add a computational study of the original reaction model to validate and extend the analytical findings. Our results show that the absence of DNA methylation reduces the bias of the system’s stationary probability distribution toward the repressed chromatin state and the temporal duration of this state’s memory. In the absence of repressive histone modifications, we also observe that the time needed to reactivate a repressed gene with an activating input is less stochastic, suggesting that repressive histone modifications specifically contribute to the highly variable latency of state reactivation. 

Epigenetic cell memory (ECM), the inheritance of gene expression patterns without changes in genetic sequence, is a critical property of multi-cellular organisms. Chromatin state, as dictated by histone covalent modifications, has recently appeared as a mediator of ECM. In this paper, we conduct a stochastic analysis of the histone modification circuit that controls chromatin state to determine key biological parameters that affect ECM. Specifically, we derive a one-dimensional Markov chain model of the circuit and analytically evaluate both the stationary probability distribution of chromatin state and the mean time to switch between active and repressed chromatin states. We then validate our analytical findings using stochastic simulations of the original higher dimensional circuit reaction model. Our analysis shows that as the speed of basal decay of histone modifications decreases compared to the speed of autocatalysis, the stationary probability distribution becomes bimodal and increasingly concentrated about the active and repressed chromatin states. Accordingly, the switching time between active and repressed chromatin states becomes larger. These results indicate that time scale separation among key constituent processes of the histone modification circuit controls ECM. 

Epigenetic cell memory allows distinct gene expression patterns to persist in different cell types despite a common genotype. Although different patterns can be maintained by the concerted action of transcription factors (TFs), it was proposed that long-term persistence hinges on chromatin state. Here, we study how the dynamics of chromatin state affect memory, and focus on a biologically motivated circuit motif, among histones and DNA modifications, that mediates the action of TFs on gene expression. Memory arises from time-scale separation among three circuit’s constituent processes: basal erasure, auto and cross-catalysis, and recruited erasure of modifications. When the two latter processes are sufficiently faster than the former, the circuit exhibits bistability and hysteresis, allowing active and repressed gene states to coexist and persist after TF stimulus removal. The duration of memory is stochastic with a mean value that increases as time-scale separation increases, but more so for the repressed state. This asymmetry stems from the cross-catalysis between repressive histone modifications and DNA methylation and is enhanced by the relatively slower decay rate of the latter. Nevertheless, TF-mediated positive autoregulation can rebalance this asymmetry and even confers robustness of active states to repressive stimuli. More generally, by wiring positively autoregulated chromatin modification circuits under time scale separation, long-term distinct gene expression patterns arise, which are also robust to failure in the regulatory links.